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</div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2503.13146">arXiv:2503.13146</a> <span> [<a href="https://arxiv.org/pdf/2503.13146">pdf</a>, <a href="https://arxiv.org/ps/2503.13146">ps</a>, <a href="https://arxiv.org/format/2503.13146">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Topology">math.GN</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Logic">math.LO</span> </div> </div> <p class="title is-5 mathjax"> On Sierpi艅ski sets, Hurewicz spaces and Hilgers functions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Marciszewski%2C+W">Witold Marciszewski</a>, <a href="/search/math?searchtype=author&query=Pol%2C+R">Roman Pol</a>, <a href="/search/math?searchtype=author&query=Zakrzewski%2C+P">Piotr Zakrzewski</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2503.13146v1-abstract-short" style="display: inline;"> The Hurewicz property is a classical generalization of $蟽$-compactness and Sierpi艅ski sets (whose existence follows from CH) are standard examples of non-$蟽$-compact Hurewicz spaces. We show, solving a problem stated by Szewczak and Tsaban, that for each Sierpi艅ski set S of cardinality at least $\mathfrak b$ there is a Hurewicz space H with $S\times H$ not Hurewicz. Some other questions in the lit… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.13146v1-abstract-full').style.display = 'inline'; document.getElementById('2503.13146v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2503.13146v1-abstract-full" style="display: none;"> The Hurewicz property is a classical generalization of $蟽$-compactness and Sierpi艅ski sets (whose existence follows from CH) are standard examples of non-$蟽$-compact Hurewicz spaces. We show, solving a problem stated by Szewczak and Tsaban, that for each Sierpi艅ski set S of cardinality at least $\mathfrak b$ there is a Hurewicz space H with $S\times H$ not Hurewicz. Some other questions in the literature concerning this topic are also answered. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.13146v1-abstract-full').style.display = 'none'; document.getElementById('2503.13146v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 54D20; 54B10; 03E17 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.18618">arXiv:2407.18618</a> <span> [<a href="https://arxiv.org/pdf/2407.18618">pdf</a>, <a href="https://arxiv.org/ps/2407.18618">ps</a>, <a href="https://arxiv.org/format/2407.18618">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Topology">math.GN</span> </div> </div> <p class="title is-5 mathjax"> Characterizing function spaces which have the property (B) of Banakh </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krupski%2C+M">Miko艂aj Krupski</a>, <a href="/search/math?searchtype=author&query=Kucharski%2C+K">Kacper Kucharski</a>, <a href="/search/math?searchtype=author&query=Marciszewski%2C+W">Witold Marciszewski</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.18618v1-abstract-short" style="display: inline;"> A topological space $Y$ has the property (B) of Banakh if there is a countable family $\{A_n:n\in \mathbb{N}\}$ of closed nowhere dense subsets of $Y$ absorbing all compact subsets of $Y$. In this note we show that the space $C_p(X)$ of continuous real-valued functions on a Tychonoff space $X$ with the topology of pointwise convergence, fails to satisfy the property (B) if and only if the space… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.18618v1-abstract-full').style.display = 'inline'; document.getElementById('2407.18618v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.18618v1-abstract-full" style="display: none;"> A topological space $Y$ has the property (B) of Banakh if there is a countable family $\{A_n:n\in \mathbb{N}\}$ of closed nowhere dense subsets of $Y$ absorbing all compact subsets of $Y$. In this note we show that the space $C_p(X)$ of continuous real-valued functions on a Tychonoff space $X$ with the topology of pointwise convergence, fails to satisfy the property (B) if and only if the space $X$ has the following property $(魏)$: every sequence of disjoint finite subsets of $X$ has a subsequence with point--finite open expansion. Additionally, we provide an analogous characterization for the compact--open topology on $C(X)$. Finally, we give examples of Tychonoff spaces $X$ whose all bounded subsets are finite, yet $X$ fails to have the property $(魏)$. This answers a question of Tkachuk. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.18618v1-abstract-full').style.display = 'none'; document.getElementById('2407.18618v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 54C35; 54E52; 54A10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2303.03809">arXiv:2303.03809</a> <span> [<a href="https://arxiv.org/pdf/2303.03809">pdf</a>, <a href="https://arxiv.org/ps/2303.03809">ps</a>, <a href="https://arxiv.org/format/2303.03809">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Topology">math.GN</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> </div> <p class="title is-5 mathjax"> On sequences of finitely supported measures related to the Josefson--Nissenzweig theorem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Marciszewski%2C+W">Witold Marciszewski</a>, <a href="/search/math?searchtype=author&query=Sobota%2C+D">Damian Sobota</a>, <a href="/search/math?searchtype=author&query=Zdomskyy%2C+L">Lyubomyr Zdomskyy</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2303.03809v2-abstract-short" style="display: inline;"> Given a Tychonoff space $X$, we call a sequence $\langle渭_n\colon n\in蠅\rangle$ of signed Borel measures on $X$ a finitely supported Josefson--Nissenzweig sequence (in short a JN-sequence) if: 1) for every $n\in蠅$ the measure $渭_n$ is a finite combination of one-point measures and $\|渭_n\|=1$, and 2) $\int_Xf\,\mathrm{d}渭_n\to0$ for every continuous function $f\in C(X)$. Our main result asserts th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.03809v2-abstract-full').style.display = 'inline'; document.getElementById('2303.03809v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2303.03809v2-abstract-full" style="display: none;"> Given a Tychonoff space $X$, we call a sequence $\langle渭_n\colon n\in蠅\rangle$ of signed Borel measures on $X$ a finitely supported Josefson--Nissenzweig sequence (in short a JN-sequence) if: 1) for every $n\in蠅$ the measure $渭_n$ is a finite combination of one-point measures and $\|渭_n\|=1$, and 2) $\int_Xf\,\mathrm{d}渭_n\to0$ for every continuous function $f\in C(X)$. Our main result asserts that if a Tychonoff space $X$ admits a JN-sequence, then there exists a JN-sequence $\langle渭_n\colon n\in蠅\rangle$ such that: i) $\mbox{supp}(渭_n)\cap\mbox{supp}(渭_k)=\emptyset$ for every $n\neq k\in蠅$, and ii) the union $\bigcup_{n\in蠅}\mbox{supp}(渭_n)$ is a discrete subset of $X$. We also prove that if a Tychonoff space $X$ carries a JN-sequence, then either there is a JN-sequence $\langle渭_n\colon n\in蠅\rangle$ on $X$ such that $|\mbox{supp}(渭_n)|=2$ for every $n\in蠅$, or for every JN-sequence $\langle渭_n\colon n\in蠅\rangle$ on $X$ we have $\lim_{n\to\infty}|\mbox{supp}(渭_n)|=\infty$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.03809v2-abstract-full').style.display = 'none'; document.getElementById('2303.03809v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 July, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 March, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">arXiv admin note: substantial text overlap with arXiv:2009.07552</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2204.01557">arXiv:2204.01557</a> <span> [<a href="https://arxiv.org/pdf/2204.01557">pdf</a>, <a href="https://arxiv.org/ps/2204.01557">ps</a>, <a href="https://arxiv.org/format/2204.01557">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="General Topology">math.GN</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Logic">math.LO</span> </div> </div> <p class="title is-5 mathjax"> The Josefson--Nissenzweig theorem and filters on $蠅$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Marciszewski%2C+W">Witold Marciszewski</a>, <a href="/search/math?searchtype=author&query=Sobota%2C+D">Damian Sobota</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2204.01557v3-abstract-short" style="display: inline;"> For a free filter $F$ on $蠅$, endow the space $N_F=蠅\cup\{p_F\}$, where $p_F\not\in蠅$, with the topology in which every element of $蠅$ is isolated whereas all open neighborhoods of $p_F$ are of the form $A\cup\{p_F\}$ for $A\in F$. Spaces of the form $N_F$ constitute the class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Jos… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2204.01557v3-abstract-full').style.display = 'inline'; document.getElementById('2204.01557v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2204.01557v3-abstract-full" style="display: none;"> For a free filter $F$ on $蠅$, endow the space $N_F=蠅\cup\{p_F\}$, where $p_F\not\in蠅$, with the topology in which every element of $蠅$ is isolated whereas all open neighborhoods of $p_F$ are of the form $A\cup\{p_F\}$ for $A\in F$. Spaces of the form $N_F$ constitute the class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson--Nissenzweig theorem from Banach space theory. We prove, e.g., that, for a filter $F$, the space $N_F$ carries a sequence $\langle渭_n\colon n\in蠅\rangle$ of normalized finitely supported signed measures such that $渭_n(f)\to 0$ for every bounded continuous real-valued function $f$ on $N_F$ if and only if $F^*\le_K\mathcal{Z}$, that is, the dual ideal $F^*$ is Kat臎tov below the asymptotic density ideal $\mathcal{Z}$. Consequently, we get that if $F^*\le_K\mathcal{Z}$, then: (1) if $X$ is a Tychonoff space and $N_F$ is homeomorphic to a subspace of $X$, then the space $C_p^*(X)$ of bounded continuous real-valued functions on $X$ contains a complemented copy of the space $c_0$ endowed with the pointwise topology, (2) if $K$ is a compact Hausdorff space and $N_F$ is homeomorphic to a subspace of $K$, then the Banach space $C(K)$ of continuous real-valued functions on $K$ is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space $K$ contains a non-trivial convergent sequence, then the space $C(K)$ is not Grothendieck. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2204.01557v3-abstract-full').style.display = 'none'; document.getElementById('2204.01557v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 April, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">30 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2107.02513">arXiv:2107.02513</a> <span> [<a href="https://arxiv.org/pdf/2107.02513">pdf</a>, <a href="https://arxiv.org/ps/2107.02513">ps</a>, <a href="https://arxiv.org/format/2107.02513">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Topology">math.GN</span> </div> </div> <p class="title is-5 mathjax"> Digging into the classes of $魏$-Corson compact spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Marciszewski%2C+W">Witold Marciszewski</a>, <a href="/search/math?searchtype=author&query=Plebanek%2C+G">Grzegorz Plebanek</a>, <a href="/search/math?searchtype=author&query=Zakrzewski%2C+K">Krzysztof Zakrzewski</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2107.02513v5-abstract-short" style="display: inline;"> For any cardinal number $魏$ and an index set $螕$, $危_魏$-product of real lines consists of elements of ${\mathbb R}^螕$ having $<魏$ nonzero coordinates. A compact space $K$ is $魏$-Corson compact if it can be embedded into such a space for some $螕$. The class of ($蠅_1$-)Corson compact spaces has been intensively studied over last decades. We discuss properties of $魏$-Corson compacta for various cardi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2107.02513v5-abstract-full').style.display = 'inline'; document.getElementById('2107.02513v5-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2107.02513v5-abstract-full" style="display: none;"> For any cardinal number $魏$ and an index set $螕$, $危_魏$-product of real lines consists of elements of ${\mathbb R}^螕$ having $<魏$ nonzero coordinates. A compact space $K$ is $魏$-Corson compact if it can be embedded into such a space for some $螕$. The class of ($蠅_1$-)Corson compact spaces has been intensively studied over last decades. We discuss properties of $魏$-Corson compacta for various cardinal numbers $魏$ as well as properties of related Boolean algebras and spaces of continuous functions. We present here a detailed discussion of the class of $蠅$-Corson compacta extending the results of Nakhmanson and Yakovlev. For $魏>蠅$, our results on $魏$-Corson compact spaces are related to the line of research originated by Kalenda and Bell and Marciszewski, and continued by Bonnet, Kubis and Todorcevic in their recent paper. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2107.02513v5-abstract-full').style.display = 'none'; document.getElementById('2107.02513v5-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 July, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 July, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">37 pages; version of February 26, 2023; the paper contains, in particular, a preliminary report from 2021</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 46A50; 54D30; 54G12 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2103.03153">arXiv:2103.03153</a> <span> [<a href="https://arxiv.org/pdf/2103.03153">pdf</a>, <a href="https://arxiv.org/ps/2103.03153">ps</a>, <a href="https://arxiv.org/format/2103.03153">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Topology">math.GN</span> </div> </div> <p class="title is-5 mathjax"> On two problems concerning Eberlein compacta </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Marciszewski%2C+W">Witold Marciszewski</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2103.03153v1-abstract-short" style="display: inline;"> We discuss two problems concerning the class Eberlein compacta, i.e., weakly compact subspaces of Banach spaces. The first one deals with preservation of some classes of scattered Eberlein compacta under continuous images. The second one concerns the known problem of the existence of nonmetrizable compact spaces without nonmetrizable zero-dimensional closed subspaces. We show that the existence of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.03153v1-abstract-full').style.display = 'inline'; document.getElementById('2103.03153v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2103.03153v1-abstract-full" style="display: none;"> We discuss two problems concerning the class Eberlein compacta, i.e., weakly compact subspaces of Banach spaces. The first one deals with preservation of some classes of scattered Eberlein compacta under continuous images. The second one concerns the known problem of the existence of nonmetrizable compact spaces without nonmetrizable zero-dimensional closed subspaces. We show that the existence of such Eberlein compacta is consistent with ZFC. We also show that it is consistent with ZFC that each Eberlein compact space of weight $> 蠅_1$ contains a nonmetrizable closed zero-dimensional subspace. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.03153v1-abstract-full').style.display = 'none'; document.getElementById('2103.03153v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 March, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 46A50; 54D30; 54G12 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2007.14723">arXiv:2007.14723</a> <span> [<a href="https://arxiv.org/pdf/2007.14723">pdf</a>, <a href="https://arxiv.org/ps/2007.14723">ps</a>, <a href="https://arxiv.org/format/2007.14723">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Topology">math.GN</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Logic">math.LO</span> </div> </div> <p class="title is-5 mathjax"> On complemented copies of the space $c_0$ in spaces $C_p(X\times Y)$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=K%C4%85kol%2C+J">Jerzy K膮kol</a>, <a href="/search/math?searchtype=author&query=Marciszewski%2C+W">Witold Marciszewski</a>, <a href="/search/math?searchtype=author&query=Sobota%2C+D">Damian Sobota</a>, <a href="/search/math?searchtype=author&query=Zdomskyy%2C+L">Lyubomyr Zdomskyy</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2007.14723v2-abstract-short" style="display: inline;"> Cembranos and Freniche proved that for every two infinite compact Hausdorff spaces $X$ and $Y$ the Banach space $C(X\times Y)$ of continuous real-valued functions on $X\times Y$ endowed with the supremum norm contains a complemented copy of the Banach space $c_{0}$. We extend this theorem to the class of $C_p$-spaces, that is, we prove that for all infinite Tychonoff spaces $X$ and $Y$ the space… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2007.14723v2-abstract-full').style.display = 'inline'; document.getElementById('2007.14723v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2007.14723v2-abstract-full" style="display: none;"> Cembranos and Freniche proved that for every two infinite compact Hausdorff spaces $X$ and $Y$ the Banach space $C(X\times Y)$ of continuous real-valued functions on $X\times Y$ endowed with the supremum norm contains a complemented copy of the Banach space $c_{0}$. We extend this theorem to the class of $C_p$-spaces, that is, we prove that for all infinite Tychonoff spaces $X$ and $Y$ the space $C_{p}(X\times Y)$ of continuous functions on $X\times Y$ endowed with the pointwise topology contains either a complemented copy of $\mathbb{R}^蠅$ or a complemented copy of the space $(c_{0})_{p}=\{(x_n)_{n\in蠅}\in \mathbb{R}^蠅\colon x_n\to 0\}$, both endowed with the product topology. We show that the latter case holds always when $X\times Y$ is pseudocompact. On the other hand, assuming the Continuum Hypothesis (or even a weaker set-theoretic assumption), we provide an example of a pseudocompact space $X$ such that $C_{p}(X\times X)$ does not contain a complemented copy of $(c_{0})_{p}$. As a corollary to the first result, we show that for all infinite Tychonoff spaces $X$ and $Y$ the space $C_{p}(X\times Y)$ is linearly homeomorphic to the space $C_{p}(X\times Y)\times\mathbb{R}$, although, as proved earlier by Marciszewski, there exists an infinite compact space $X$ such that $C_{p}(X)$ cannot be mapped onto $C_{p}(X)\times\mathbb{R}$ by a continuous linear surjection. This provides a positive answer to a problem of Arkhangel'ski for spaces of the form $C_p(X\times Y)$. Another corollary asserts that for every infinite Tychonoff spaces $X$ and $Y$ the space $C_{k}(X\times Y)$ of continuous functions on $X\times Y$ endowed with the compact-open topology admits a quotient map onto a space isomorphic to one of the following three spaces: $\mathbb{R}^蠅$, $(c_{0})_{p}$ or $c_{0}$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2007.14723v2-abstract-full').style.display = 'none'; document.getElementById('2007.14723v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 June, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 July, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">30 pages, comments are welcome</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2002.07423">arXiv:2002.07423</a> <span> [<a href="https://arxiv.org/pdf/2002.07423">pdf</a>, <a href="https://arxiv.org/ps/2002.07423">ps</a>, <a href="https://arxiv.org/format/2002.07423">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Topology">math.GN</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1090/proc/15271">10.1090/proc/15271 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A countable dense homogeneous topological vector space is a Baire space </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Dobrowolski%2C+T">Tadeusz Dobrowolski</a>, <a href="/search/math?searchtype=author&query=Krupski%2C+M">Miko艂aj Krupski</a>, <a href="/search/math?searchtype=author&query=Marciszewski%2C+W">Witold Marciszewski</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2002.07423v2-abstract-short" style="display: inline;"> We prove that every homogeneous countable dense homogeneous topological space containing a copy of the Cantor set is a Baire space. In particular, every countable dense homogeneous topological vector space is a Baire space. It follows that, for any nondiscrete metrizable space $X$, the function space $C_p(X)$ is not countable dense homogeneous. This answers a question posed recently by R. Hern谩nde… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.07423v2-abstract-full').style.display = 'inline'; document.getElementById('2002.07423v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2002.07423v2-abstract-full" style="display: none;"> We prove that every homogeneous countable dense homogeneous topological space containing a copy of the Cantor set is a Baire space. In particular, every countable dense homogeneous topological vector space is a Baire space. It follows that, for any nondiscrete metrizable space $X$, the function space $C_p(X)$ is not countable dense homogeneous. This answers a question posed recently by R. Hern谩ndez-Guti茅rrez. We also conclude that, for any infinite dimensional Banach space $E$ (dual Banach space $E^\ast$), the space $E$ equipped with the weak topology ($E^\ast$ with the weak$^\ast$ topology) is not countable dense homogeneous. We generalize some results of Hru拧谩k, Zamora Avil茅s, and Hern谩ndez-Guti茅rrez concerning countable dense homogeneous products. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.07423v2-abstract-full').style.display = 'none'; document.getElementById('2002.07423v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 April, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 February, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">slightly modified and expanded version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 54C35; 54E52; 46A03; Secondary: 22A05 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Proc. Amer. Math. Soc. 149 (2021), 1773-1789 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1910.07273">arXiv:1910.07273</a> <span> [<a href="https://arxiv.org/pdf/1910.07273">pdf</a>, <a href="https://arxiv.org/ps/1910.07273">ps</a>, <a href="https://arxiv.org/format/1910.07273">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> </div> <p class="title is-5 mathjax"> Sailing over three problems of Koszmider </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=S%C3%A1nchez%2C+F+C">F茅lix Cabello S谩nchez</a>, <a href="/search/math?searchtype=author&query=Castillo%2C+J+M+F">Jes煤s M. F. Castillo</a>, <a href="/search/math?searchtype=author&query=Marciszewski%2C+W">Witold Marciszewski</a>, <a href="/search/math?searchtype=author&query=Plebanek%2C+G">Grzegorz Plebanek</a>, <a href="/search/math?searchtype=author&query=Salguero-Alarc%C3%B3n%2C+A">Alberto Salguero-Alarc贸n</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1910.07273v2-abstract-short" style="display: inline;"> We discuss three problems of Koszmider on the structure of the spaces of continuous functions on the Stone compact $K_{\mathcal A}$ generated by an almost disjoint family $\mathcal A$ of infinite subsets of $蠅$ -- we present a solution to two problems and develop a previous results of Marciszewski and Pol answering the third one. We will show, in particular, that assuming Martin's axiom the space… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1910.07273v2-abstract-full').style.display = 'inline'; document.getElementById('1910.07273v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1910.07273v2-abstract-full" style="display: none;"> We discuss three problems of Koszmider on the structure of the spaces of continuous functions on the Stone compact $K_{\mathcal A}$ generated by an almost disjoint family $\mathcal A$ of infinite subsets of $蠅$ -- we present a solution to two problems and develop a previous results of Marciszewski and Pol answering the third one. We will show, in particular, that assuming Martin's axiom the space $C(K_{\mathcal A})$ is uniquely determined up to isomorphism by the cardinality of $\mathcal A$ whenever $|{\mathcal A}|<{\mathfrak c}$, while there are $2^{\mathfrak c}$ nonisomorphic spaces $C(K_{\mathcal A})$ with $|{\mathcal A}|= {\mathfrak c}$. We also investigate Koszmider's problems in the context of the class of separable Rosenthal compacta and indicate the meaning of our results in the language of twisted sums of $c_0$ and some $C(K)$ spaces. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1910.07273v2-abstract-full').style.display = 'none'; document.getElementById('1910.07273v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 April, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 October, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages; the final version, accepted for publication</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 46E15; 03E50; 54G12 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1902.07783">arXiv:1902.07783</a> <span> [<a href="https://arxiv.org/pdf/1902.07783">pdf</a>, <a href="https://arxiv.org/ps/1902.07783">ps</a>, <a href="https://arxiv.org/format/1902.07783">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> </div> <p class="title is-5 mathjax"> Twisted sums of $c_0$ and $C(K)$-spaces: A solution to the CCKY problem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Avil%C3%A9s%2C+A">Antonio Avil茅s</a>, <a href="/search/math?searchtype=author&query=Marciszewski%2C+W">Witold Marciszewski</a>, <a href="/search/math?searchtype=author&query=Plebanek%2C+G">Grzegorz Plebanek</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1902.07783v3-abstract-short" style="display: inline;"> We consider the class of Banach space $Y$ for which $c_0$ admits a nontrivial twisted sum with $Y$. We present a characterization of such space $Y$ in terms of properties of the $weak^\ast$ topology on $Y^\ast$. We prove that under the continuum hypothesis $c_0$ has a nontrivial twisted sum with every space of the form $Y=C(K)$, where $K$ is compact and not metrizable. This gives a consistent posi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1902.07783v3-abstract-full').style.display = 'inline'; document.getElementById('1902.07783v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1902.07783v3-abstract-full" style="display: none;"> We consider the class of Banach space $Y$ for which $c_0$ admits a nontrivial twisted sum with $Y$. We present a characterization of such space $Y$ in terms of properties of the $weak^\ast$ topology on $Y^\ast$. We prove that under the continuum hypothesis $c_0$ has a nontrivial twisted sum with every space of the form $Y=C(K)$, where $K$ is compact and not metrizable. This gives a consistent positive solution to a problem posed by Cabello, Castillo, Kalton and Yost. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1902.07783v3-abstract-full').style.display = 'none'; document.getElementById('1902.07783v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 April, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 February, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Final version (the previous title `Twisting $c_0$ around nonseparable Banach spaces'), 32 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 46B25; 46B26; 46E15 (Primary); 03E35; 54C55; 54D40 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1710.10510">arXiv:1710.10510</a> <span> [<a href="https://arxiv.org/pdf/1710.10510">pdf</a>, <a href="https://arxiv.org/ps/1710.10510">ps</a>, <a href="https://arxiv.org/format/1710.10510">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Topology">math.GN</span> </div> </div> <p class="title is-5 mathjax"> On uniformly continuous maps between function spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gorak%2C+R">Rafal Gorak</a>, <a href="/search/math?searchtype=author&query=Krupski%2C+M">Mikolaj Krupski</a>, <a href="/search/math?searchtype=author&query=Marciszewski%2C+W">Witold Marciszewski</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1710.10510v1-abstract-short" style="display: inline;"> In this paper we develop a technique of constructing uni- formly continuous maps between function spaces Cp(X) endowed with the pointwise topology. We prove that if a space X is compact metrizable and strongly countable-dimensional, then there exists a uniformly contin- uous surjection from Cp([0,1]) onto Cp(X). We provide a partial result concerning the reverse implication. We also show that, for… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1710.10510v1-abstract-full').style.display = 'inline'; document.getElementById('1710.10510v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1710.10510v1-abstract-full" style="display: none;"> In this paper we develop a technique of constructing uni- formly continuous maps between function spaces Cp(X) endowed with the pointwise topology. We prove that if a space X is compact metrizable and strongly countable-dimensional, then there exists a uniformly contin- uous surjection from Cp([0,1]) onto Cp(X). We provide a partial result concerning the reverse implication. We also show that, for every infinite Polish zero-dimensional space X, the spaces Cp(X) and Cp(X) x Cp(X) are uniformly homeomorphic. This partially answers two questions posed by Krupski and Marciszewski. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1710.10510v1-abstract-full').style.display = 'none'; document.getElementById('1710.10510v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 October, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 54C35 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1703.02139">arXiv:1703.02139</a> <span> [<a href="https://arxiv.org/pdf/1703.02139">pdf</a>, <a href="https://arxiv.org/ps/1703.02139">ps</a>, <a href="https://arxiv.org/format/1703.02139">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> </div> <p class="title is-5 mathjax"> Extension operators and twisted sums of $c_0$ and $C(K)$ spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Marciszewski%2C+W">Witold Marciszewski</a>, <a href="/search/math?searchtype=author&query=Plebanek%2C+G">Grzegorz Plebanek</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1703.02139v2-abstract-short" style="display: inline;"> We investigate the following problem posed by Cabello Sanch茅z, Castillo, Kalton, and Yost: Let $K$ be a nonmetrizable compact space. Does there exist a nontrivial twisted sum of $c_0$ and $C(K)$, i.e., does there exist a Banach space $X$ containing a non-complemented copy $Z$ of $c_0$ such that the quotient space $X/Z$ is isomorphic to $C(K)$? Using additional set-theoretic assumptions we give… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1703.02139v2-abstract-full').style.display = 'inline'; document.getElementById('1703.02139v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1703.02139v2-abstract-full" style="display: none;"> We investigate the following problem posed by Cabello Sanch茅z, Castillo, Kalton, and Yost: Let $K$ be a nonmetrizable compact space. Does there exist a nontrivial twisted sum of $c_0$ and $C(K)$, i.e., does there exist a Banach space $X$ containing a non-complemented copy $Z$ of $c_0$ such that the quotient space $X/Z$ is isomorphic to $C(K)$? Using additional set-theoretic assumptions we give the first examples of compact spaces $K$ providing a negative answer to this question. We show that under Martin's axiom and the negation of the continuum hypothesis, if either $K$ is the Cantor cube $2^{蠅_1}$ or $K$ is a separable scattered compact space of height $3$ and weight $蠅_1$, then every twisted sum of $c_0$ and $C(K)$ is trivial. We also construct nontrivial twisted sums of $c_0$ and $C(K)$ for $K$ belonging to several classes of compacta. Our main tool is an investigation of pairs of compact spaces $K\subseteq L$ which do not admit an extension operator $C(K)\to C(L)$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1703.02139v2-abstract-full').style.display = 'none'; document.getElementById('1703.02139v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 August, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 March, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">34 pages, revised version of August 11, 2017</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 46B25; 46B26; 46E15 (Primary); 03E35; 54C55; 54D40 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1608.03883">arXiv:1608.03883</a> <span> [<a href="https://arxiv.org/pdf/1608.03883">pdf</a>, <a href="https://arxiv.org/ps/1608.03883">ps</a>, <a href="https://arxiv.org/format/1608.03883">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Topology">math.GN</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.jmaa.2017.02.066">10.1016/j.jmaa.2017.02.066 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the weak and pointwise topologies in function spaces II </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krupski%2C+M">Miko艂aj Krupski</a>, <a href="/search/math?searchtype=author&query=Marciszewski%2C+W">Witold Marciszewski</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1608.03883v2-abstract-short" style="display: inline;"> For a compact space $K$ we denote by $C_w(K)$ ($C_p(K)$) the space of continuous real-valued functions on $K$ endowed with the weak (pointwise) topology. In this paper we discuss the following basic question which seems to be open: Let $K$ and $L$ be infinite compact spaces. Can it happen that $C_w(K)$ and $C_p(L)$ are homeomorphic? M. Krupski proved that the above problem has a negative answer… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1608.03883v2-abstract-full').style.display = 'inline'; document.getElementById('1608.03883v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1608.03883v2-abstract-full" style="display: none;"> For a compact space $K$ we denote by $C_w(K)$ ($C_p(K)$) the space of continuous real-valued functions on $K$ endowed with the weak (pointwise) topology. In this paper we discuss the following basic question which seems to be open: Let $K$ and $L$ be infinite compact spaces. Can it happen that $C_w(K)$ and $C_p(L)$ are homeomorphic? M. Krupski proved that the above problem has a negative answer when $K=L$ and $K$ is finite-dimensional and metrizable. We extend this result to the class of finite-dimensional Valdivia compact spaces $K$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1608.03883v2-abstract-full').style.display = 'none'; document.getElementById('1608.03883v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 February, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 August, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 46E10; 54C35 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Math. Anal. Appl. 452 (2017), no. 1, 646--658 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1503.04229">arXiv:1503.04229</a> <span> [<a href="https://arxiv.org/pdf/1503.04229">pdf</a>, <a href="https://arxiv.org/ps/1503.04229">ps</a>, <a href="https://arxiv.org/format/1503.04229">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Topology">math.GN</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s11856-016-1373-y">10.1007/s11856-016-1373-y <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A metrizable $X$ with $C_p(X)$ not homeomorphic to $C_p(X)\times C_p(X)$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krupski%2C+M">Miko艂aj Krupski</a>, <a href="/search/math?searchtype=author&query=Marciszewski%2C+W">Witold Marciszewski</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1503.04229v2-abstract-short" style="display: inline;"> We give an example of an infinite metrizable space $X$ such that the space $C_p(X)$, of continuous real-valued function on $X$ endowed with the pointwise topology, is not homeomorphic to its own square $C_p(X)\times C_p(X)$. The space $X$ is a zero-dimensional subspace of the real line. Our result answers a long-standing open question in the theory of function spaces posed by A.V. Arhangel'skii. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1503.04229v2-abstract-full" style="display: none;"> We give an example of an infinite metrizable space $X$ such that the space $C_p(X)$, of continuous real-valued function on $X$ endowed with the pointwise topology, is not homeomorphic to its own square $C_p(X)\times C_p(X)$. The space $X$ is a zero-dimensional subspace of the real line. Our result answers a long-standing open question in the theory of function spaces posed by A.V. Arhangel'skii. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1503.04229v2-abstract-full').style.display = 'none'; document.getElementById('1503.04229v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 July, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 March, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 46E10; 54C35 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Israel J. Math. 214 (2016), no. 1, 245--258 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1502.01875">arXiv:1502.01875</a> <span> [<a href="https://arxiv.org/pdf/1502.01875">pdf</a>, <a href="https://arxiv.org/ps/1502.01875">ps</a>, <a href="https://arxiv.org/format/1502.01875">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="General Topology">math.GN</span> </div> </div> <p class="title is-5 mathjax"> Extension operators on balls and on spaces of finite sets </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Avil%C3%A9s%2C+A">Antonio Avil茅s</a>, <a href="/search/math?searchtype=author&query=Marciszewski%2C+W">Witold Marciszewski</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1502.01875v1-abstract-short" style="display: inline;"> We study extension operators between spaces $蟽_n(2^X)$ of subsets of $X$ of cardinality at most $n$. As an application, we show that if $B_H$ is the unit ball of a nonseparable Hilbert space $H$, equipped with the weak topology, then, for any $0<位<渭$, there is no extension operator $T: C(位B_H)\to C(渭B_H)$. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1502.01875v1-abstract-full" style="display: none;"> We study extension operators between spaces $蟽_n(2^X)$ of subsets of $X$ of cardinality at most $n$. As an application, we show that if $B_H$ is the unit ball of a nonseparable Hilbert space $H$, equipped with the weak topology, then, for any $0<位<渭$, there is no extension operator $T: C(位B_H)\to C(渭B_H)$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1502.01875v1-abstract-full').style.display = 'none'; document.getElementById('1502.01875v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 February, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 46B26; 46E15; 54C35; 54H05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1412.1748">arXiv:1412.1748</a> <span> [<a href="https://arxiv.org/pdf/1412.1748">pdf</a>, <a href="https://arxiv.org/ps/1412.1748">ps</a>, <a href="https://arxiv.org/format/1412.1748">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.jmaa.2015.07.037">10.1016/j.jmaa.2015.07.037 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Networks for the weak topology of Banach and Fr茅chet spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gabriyelyan%2C+S">S. Gabriyelyan</a>, <a href="/search/math?searchtype=author&query=Kcakol%2C+J">J. Kcakol</a>, <a href="/search/math?searchtype=author&query=Kubi%C5%9B%2C+W">W. Kubi艣</a>, <a href="/search/math?searchtype=author&query=Marciszewski%2C+W">W. Marciszewski</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1412.1748v1-abstract-short" style="display: inline;"> We start the systematic study of Fr茅chet spaces which are $\aleph$-spaces in the weak topology. A topological space $X$ is an $\aleph_0$-space or an $\aleph$-space if $X$ has a countable $k$-network or a $蟽$-locally finite $k$-network, respectively. We are motivated by the following result of Corson (1966): If the space $C_{c}(X)$ of continuous real-valued functions on a Tychonoff space $X$ endowe… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1412.1748v1-abstract-full').style.display = 'inline'; document.getElementById('1412.1748v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1412.1748v1-abstract-full" style="display: none;"> We start the systematic study of Fr茅chet spaces which are $\aleph$-spaces in the weak topology. A topological space $X$ is an $\aleph_0$-space or an $\aleph$-space if $X$ has a countable $k$-network or a $蟽$-locally finite $k$-network, respectively. We are motivated by the following result of Corson (1966): If the space $C_{c}(X)$ of continuous real-valued functions on a Tychonoff space $X$ endowed with the compact-open topology is a Banach space, then $C_{c}(X)$ endowed with the weak topology is an $\aleph_0$-space if and only if $X$ is countable. We extend Corson's result as follows: If the space $E:=C_{c}(X)$ is a Fr茅chet lcs, then $E$ endowed with its weak topology $蟽(E,E')$ is an $\aleph$-space if and only if $(E,蟽(E,E'))$ is an $\aleph_0$-space if and only if $X$ is countable. We obtain a necessary and some sufficient conditions on a Fr茅chet lcs to be an $\aleph$-space in the weak topology. We prove that a reflexive Fr茅chet lcs $E$ in the weak topology $蟽(E,E')$ is an $\aleph$-space if and only if $(E,蟽(E,E'))$ is an $\aleph_0$-space if and only if $E$ is separable. We show however that the nonseparable Banach space $\ell_{1}(\mathbb{R})$ with the weak topology is an $\aleph$-space. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1412.1748v1-abstract-full').style.display = 'none'; document.getElementById('1412.1748v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 December, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary 46A03; 54H11; Secondary 22A05; 54C35 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Math. Anal. Appl. 432 (2015), no. 2, 1183--1199 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1309.1908">arXiv:1309.1908</a> <span> [<a href="https://arxiv.org/pdf/1309.1908">pdf</a>, <a href="https://arxiv.org/ps/1309.1908">ps</a>, <a href="https://arxiv.org/format/1309.1908">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> </div> <p class="title is-5 mathjax"> On Borel structures in the Banach space C(尾蠅) </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Marciszewski%2C+W">Witold Marciszewski</a>, <a href="/search/math?searchtype=author&query=Plebanek%2C+G">Grzegorz Plebanek</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1309.1908v1-abstract-short" style="display: inline;"> M. Talagrand showed that, for the Cech-Stone compactification 尾蠅 of the space of natural numbers, the norm and the weak topology generate different Borel structures in the Banach space C(尾蠅). We prove that the Borel structures in C(尾蠅) generated by the weak and the pointwise topology are also different. We also show that in C(蠅*), where 蠅*=尾蠅- 蠅, there is no countable family of pointwise Borel s… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1309.1908v1-abstract-full').style.display = 'inline'; document.getElementById('1309.1908v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1309.1908v1-abstract-full" style="display: none;"> M. Talagrand showed that, for the Cech-Stone compactification 尾蠅 of the space of natural numbers, the norm and the weak topology generate different Borel structures in the Banach space C(尾蠅). We prove that the Borel structures in C(尾蠅) generated by the weak and the pointwise topology are also different. We also show that in C(蠅*), where 蠅*=尾蠅- 蠅, there is no countable family of pointwise Borel sets separating functions from C(蠅*). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1309.1908v1-abstract-full').style.display = 'none'; document.getElementById('1309.1908v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 September, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2013. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">14 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 46B26; 46E15; 54C35; 54H05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1207.3722">arXiv:1207.3722</a> <span> [<a href="https://arxiv.org/pdf/1207.3722">pdf</a>, <a href="https://arxiv.org/ps/1207.3722">ps</a>, <a href="https://arxiv.org/format/1207.3722">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.4064/cm128-2-4">10.4064/cm128-2-4 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Some remarks on universality properties of $\ell_\infty / c_0$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krupski%2C+M">Mikolaj Krupski</a>, <a href="/search/math?searchtype=author&query=Marciszewski%2C+W">Witold Marciszewski</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1207.3722v1-abstract-short" style="display: inline;"> We prove that if continuum is not a Kunen cardinal, then there is a uniform Eberlein compact space $K$ such that the Banach space $C(K)$ does not embed isometrically into $\ell_\infty/c_0$. We prove a similar result for isomorphic embeddings. We also construct a consistent example of a uniform Eberlein compactum whose space of continuous functions embeds isomorphically into $\ell_\infty/c_0$, but… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1207.3722v1-abstract-full').style.display = 'inline'; document.getElementById('1207.3722v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1207.3722v1-abstract-full" style="display: none;"> We prove that if continuum is not a Kunen cardinal, then there is a uniform Eberlein compact space $K$ such that the Banach space $C(K)$ does not embed isometrically into $\ell_\infty/c_0$. We prove a similar result for isomorphic embeddings. We also construct a consistent example of a uniform Eberlein compactum whose space of continuous functions embeds isomorphically into $\ell_\infty/c_0$, but fails to embed isometrically. As far as we know it is the first example of this kind. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1207.3722v1-abstract-full').style.display = 'none'; document.getElementById('1207.3722v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 July, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 46B26; 46E15 (Primary) 03E75 (Secondary) </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Colloq. Math. 128 (2012), no. 2, 187--195 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1104.2639">arXiv:1104.2639</a> <span> [<a href="https://arxiv.org/pdf/1104.2639">pdf</a>, <a href="https://arxiv.org/ps/1104.2639">ps</a>, <a href="https://arxiv.org/format/1104.2639">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> </div> <p class="title is-5 mathjax"> On measures on Rosenthal compacta </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Marciszewski%2C+W">Witold Marciszewski</a>, <a href="/search/math?searchtype=author&query=Plebanek%2C+G">Grzegorz Plebanek</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1104.2639v1-abstract-short" style="display: inline;"> We show that if K is Rosenthal compact which can be represented by functions with countably many discontinuities then every Radon measure on K is countably determined. We also present an alternative proof of the result stating that every Radon measure on an arbitrary Rosenthal compactum is of countable type. Our approach is based on some caliber-type properties of measures, parameterized by separa… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1104.2639v1-abstract-full').style.display = 'inline'; document.getElementById('1104.2639v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1104.2639v1-abstract-full" style="display: none;"> We show that if K is Rosenthal compact which can be represented by functions with countably many discontinuities then every Radon measure on K is countably determined. We also present an alternative proof of the result stating that every Radon measure on an arbitrary Rosenthal compactum is of countable type. Our approach is based on some caliber-type properties of measures, parameterized by separable metrizable spaces. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1104.2639v1-abstract-full').style.display = 'none'; document.getElementById('1104.2639v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 April, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2011. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">14 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 28C15; 46A50 (Primary) 28A60; 54C35 (Secondary) </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul class="nav-spaced"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 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