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To link to an nLab page but show a different link text, do the following: [[ name of page | link text to show ]].</li> <li style="font-size: 0.8em">LaTeX can be used inside single dollar signs (inline) or double dollar signs or \[ and \], as usual. </li> <li style="font-size: 0.8em">To create a table of contents, add \tableofcontents on its own line.</li> <li style="font-size: 0.8em">For a theorem or proof, use \begin{theorem} \end{theorem} as you would in LaTeX. Labelling and referencing is exactly as in LaTeX, with use of \label and \ref. The full list of supported environments can be found in the <a href="/nlab/show/HowTo#DefinitionTheoremProofEnvironments">HowTo</a>. </li> <li style="font-size: 0.8em">Tikz can be used for figures almost exactly as in LaTeX. Similarly, tikz-cd and xymatrix can be used for commutative diagrams. See the <a href="/nlab/show/HowTo#diagrams">HowTo</a>.</li> <li style="font-size: 0.8em">As an alternative to the Markdown syntax for sections (headings), one can use the usual LaTeX syntax \section, \subsection, etc.</li> <li style="font-size: 0.8em">For further help, see the <a href="/nlab/show/HowTo">HowTo</a>, or you are very welcome to ask at the <a href="https://nforum.ncatlab.org/">nForum</a>.</li> </ol> </div> <form accept-charset="utf-8" action="/nlab/save/Top" id="editForm" method="post"> <div style="display: none;"> <input name="see_if_human" id="see_if_human" style="tabindex: -1; autocomplete: off"/> </div> <div> <textarea name="content" id="content" style="height: 45em; width: 70%;">+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Topology +--{: .hide} [[!include topology - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Definition **Top** denotes the [[category]] whose [[objects]] are [[topological spaces]] and whose [[morphisms]] are [[continuous functions]] between them. Its [[isomorphisms]] are the [[homeomorphisms]]. For exposition see _[[Introduction to Topology -- 1|Introduction to point-set topology]]_. Often one considers (sometimes by default) [[subcategories]] of [[nice topological spaces]] such as [[compactly generated topological spaces]], notably because these are [[cartesian closed category|cartesian closed]]. There other other [[convenient categories of topological spaces]]. With any one such choice understood, it is often useful to regard it as "the" category of topological spaces. The [[homotopy category]] of $Top$ given by its [[localization]] at the [[weak homotopy equivalences]] is the [[classical homotopy category]] [[Ho(Top)]]. This is the central object of study in [[homotopy theory]], see also at _[[classical model structure on topological spaces]]_. The [[simplicial localization]] of [[Top]] at the [[weak homotopy equivalences]] is the archetypical [[(∞,1)-category]], [[equivalence of (infinity,1)-categories|equivalent]] to [[∞Grpd]] (see at _[[homotopy hypothesis]]_). ## Properties ### Universal constructions {#UniversalConstructions} We discuss [[universal constructions]] in [[Top]], such as [[limits]]/[[colimits]], etc. The following definition suggests that universal constructions be seen in the context of $Top$ as a [[topological concrete category]] (see Proposition \ref{topcat} below). $\,$ [[!include universal constructions of topological spaces -- table]] $\,$ +-- {: .num_defn #InitialAndFinalTopologies} ###### Definition **([[weak topology]] and [[strong topology]])** Let $\{X_i = (S_i,\tau_i) \in Top\}_{i \in I}$ be a [[class]] of [[topological spaces]], and let $S \in Set$ be a bare [[set]]. Then * For $\{S \stackrel{f_i}{\to} S_i \}_{i \in I}$ a set of [[functions]] out of $S$, the _[[initial topology]]_ $\tau_{initial}(\{f_i\}_{i \in I})$ is the topology on $S$ with the [[minimum]] collection of [[open subsets]] such that all $f_i \colon (S,\tau_{initial}(\{f_i\}_{i \in I}))\to X_i$ are [[continuous function|continuous]]. * For $\{S_i \stackrel{f_i}{\to} S\}_{i \in I}$ a set of [[functions]] into $S$, the _[[final topology]]_ $\tau_{final}(\{f_i\}_{i \in I})$ is the topology on $S$ with the [[maximum]] collection of [[open subsets]] such that all $f_i \colon X_i \to (S,\tau_{final}(\{f_i\}_{i \in I}))$ are [[continuous function|continuous]]. =-- +-- {: .num_example #TopologicalSubspace} ###### Example For $X$ a single topological space, and $\iota_S \colon S \hookrightarrow U(X)$ a subset of its underlying set, then the initial topology $\tau_{intial}(\iota_S)$, def. \ref{InitialAndFinalTopologies}, is the [[subspace topology]], making $$ \iota_S \;\colon\; (S, \tau_{initial}(\iota_S)) \hookrightarrow X $$ a [[topological subspace]] inclusion. =-- +-- {: .num_example #QuotientTopology} ###### Example Conversely, for $p_S \colon U(X) \longrightarrow S$ an [[epimorphism]], then the final topology $\tau_{final}(p_S)$ on $S$ is the _[[quotient topology]]_. =-- +-- {: .num_prop #DescriptionOfLimitsAndColimitsInTop} ###### Proposition Let $I$ be a [[small category]] and let $X_\bullet \colon I \longrightarrow Top$ be an $I$-[[diagram]] in [[Top]] (a [[functor]] from $I$ to $Top$), with components denoted $X_i = (S_i, \tau_i)$, where $S_i \in Set$ and $\tau_i$ a topology on $S_i$. Then: 1. The [[limit]] of $X_\bullet$ exists and is given by [[generalized the|the]] topological space whose underlying set is [[generalized the|the]] limit in [[Set]] of the underlying sets in the diagram, and whose topology is the [[initial topology]], def. \ref{InitialAndFinalTopologies}, for the functions $p_i$ which are the limiting [[cone]] components: $$ \array{ && \underset{\longleftarrow}{\lim}_{i \in I} S_i \\ & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p_j}} \\ S_i && \underset{}{\longrightarrow} && S_j } \,. $$ Hence $$ \underset{\longleftarrow}{\lim}_{i \in I} X_i \simeq \left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right) $$ 1. The [[colimit]] of $X_\bullet$ exists and is the topological space whose underlying set is the colimit in [[Set]] of the underlying diagram of sets, and whose topology is the [[final topology]], def. \ref{InitialAndFinalTopologies} for the component maps $\iota_i$ of the colimiting [[cocone]] $$ \array{ S_i && \longrightarrow && S_j \\ & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}} \\ && \underset{\longrightarrow}{\lim}_{i \in I} S_i } \,. $$ Hence $$ \underset{\longrightarrow}{\lim}_{i \in I} X_i \simeq \left(\underset{\longrightarrow}{\lim}_{i \in I} S_i,\; \tau_{final}(\{\iota_i\}_{i \in I})\right) $$ =-- (e.g. [Bourbaki 71, section I.4](#Bourbaki71)) +-- {: .proof} ###### Proof The required [[universal property]] of $\left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right)$ is immediate: for $$ \array{ && (S,\tau) \\ & {}^{\mathllap{f_i}}\swarrow && \searrow^{\mathrlap{f_j}} \\ X_i && \underset{}{\longrightarrow} && X_j } $$ any [[cone]] over the diagram, then by construction there is a unique function of underlying sets $S \longrightarrow \underset{\longleftarrow}{\lim}_{i \in I} S_i$ making the required diagrams commute, and so all that is required is that this unique function is always [[continuous function|continuous]]. But this is precisely what the initial topology ensures. The case of the colimit is [[formal dual|formally dual]]. =-- +-- {: .num_example #PointTopologicalSpaceAsEmptyLimit} ###### Example The limit over the empty diagram in $Top$ is the [[point]] $\ast$ with its unique topology. =-- +-- {: .num_example #DisjointUnionOfTopologicalSpacesIsCoproduct} ###### Example For $\{X_i\}_{i \in I}$ a set of topological spaces, their [[coproduct]] $\underset{i \in I}{\sqcup} X_i \in Top$ is their _[[disjoint union]]_. =-- In particular: +-- {: .num_example #DiscreteTopologicalSpaceAsCoproduct} ###### Example For $S \in Set$, the $S$-indexed [[coproduct]] of the point, $\underset{s \in S}{\coprod}\ast $, is the set $S$ itself equipped with the [[final topology]], hence is the [[discrete topological space]] on $S$. =-- +-- {: .num_example #ProductTopologicalSpace} ###### Example For $\{X_i\}_{i \in I}$ a set of topological spaces, their [[product]] $\underset{i \in I}{\prod} X_i \in Top$ is the [[Cartesian product]] of the underlying sets equipped with the _[[product topology]]_, also called the _[[Tychonoff product]]_. In the case that $S$ is a [[finite set]], such as for binary product spaces $X \times Y$, then a [[basis for a topology|sub-basis]] for the product topology is given by the [[Cartesian products]] of the open subsets of (a basis for) each factor space. =-- +-- {: .num_example #EqualizerInTop} ###### Example The [[equalizer]] of two [[continuous functions]] $f, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y$ in $Top$ is the equalizer of the underlying functions of sets $$ eq(f,g) \hookrightarrow S_X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} S_Y $$ (hence the largets subset of $S_X$ on which both functions coincide) and equipped with the [[subspace topology]], example \ref{TopologicalSubspace}. =-- +-- {: .num_example #CoequalizerInTop} ###### Example The [[coequalizer]] of two [[continuous functions]] $f, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y$ in $Top$ is the coequalizer of the underlying functions of sets $$ S_X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} S_Y \longrightarrow coeq(f,g) $$ (hence the [[quotient set]] by the [[equivalence relation]] generated by $f(x) \sim g(x)$ for all $x \in X$) and equipped with the [[quotient topology]], example \ref{QuotientTopology}. =-- +-- {: .num_example #PushoutInTop} ###### Example For $$ \array{ A &\overset{g}{\longrightarrow}& Y \\ {}^{\mathllap{f}}\downarrow \\ X } $$ two [[continuous functions]] out of the same [[domain]], then the [[colimit]] under this diagram is also called the _[[pushout]]_, denoted $$ \array{ A &\overset{g}{\longrightarrow}& Y \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{g_\ast f}} \\ X &\longrightarrow& X \sqcup_A Y \,. } \,. $$ (Here $g_\ast f$ is also called the pushout of $f$, or the _[[base change|cobase change]]_ of $f$ along $g$.) If $g$ is an inclusion, one also write $X \cup_f Y$ and calls this the _[[attaching space]]_. <div style="float:left;margin:0 10px 10px 0;"><img src="http://ncatlab.org/nlab/files/AttachingSpace.jpg" width="450"></div> By example \ref{CoequalizerInTop} the [[pushout]]/[[attaching space]] is the [[quotient topological space]] $$ X \sqcup_A Y \simeq (X\sqcup Y)/\sim $$ of the [[disjoint union]] of $X$ and $Y$ subject to the [[equivalence relation]] which identifies a point in $X$ with a point in $Y$ if they have the same pre-image in $A$. (graphics from [Aguilar-Gitler-Prieto 02](#AguilarGitlerPrieto02)) =-- +-- {: .num_example #TopologicalnSphereIsPushoutOfBoundaryOfnBallInclusionAlongItself} ###### Example As an important special case of example \ref{PushoutInTop}, let $$ i_n \colon S^{n-1}\longrightarrow D^n $$ be the canonical inclusion of the standard [[n-sphere|(n-1)-sphere]] as the [[boundary]] of the standard [[n-disk]] (both regarded as [[topological spaces]] with their [[subspace topology]] as subspaces of the [[Cartesian space]] $\mathbb{R}^n$). <div style="float:left;margin:0 10px 10px 0;"> <img src="http://ncatlab.org/nlab/files/GluingHemispheres.jpg" width="400"></div> Then the colimit in [[Top]] under the diagram, i.e. the [[pushout]] of $i_n$ along itself, $$ \left\{ D^n \overset{i_n}{\longleftarrow} S^{n-1} \overset{i_n}{\longrightarrow} D^n \right\} \,, $$ is the [[n-sphere]] $S^n$: $$ \array{ S^{n-1} &\overset{i_n}{\longrightarrow}& D^n \\ {}^{\mathllap{i_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& S^n } \,. $$ (graphics from Ueno-Shiga-Morita 95) =-- +-- {: .num_example #ClosedSubspacesGluing} ###### Example **([[union]] of two [[open subset|open]] or two [[closed subset|closed]] [[subspaces]] is [[pushout]])** Let $X$ be a [[topological space]] and let $A,B \subset X$ be [[subspaces]] such that 1. $A,B \subset X$ are both [[open subsets]] or are both [[closed subsets]]; 1. they constitute a [[cover]]: $X = A \cup B$ Write $i_A \colon A \to X$ and $i_B \colon B \to X$ for the corresponding inclusion [[continuous functions]]. Then the [[commuting square]] $$ \array{ A \cap B &\longrightarrow& A \\ \downarrow && \downarrow^{\mathrlap{i_A}} \\ B &\underset{i_B}{\longrightarrow}& X } $$ is a [[pushout]] square in $Top$ (example \ref{PushoutInTop}). By the [[universal property]] of the [[pushout]] this means in particular that for $Y$ any [[topological space]] then a function of underlying sets $$ f \;\colon\; X \longrightarrow Y $$ is a [[continuous function]] as soon as its two restrictions $$ f\vert_A \;\colon\; A \longrightarrow Y \phantom{AAAA} f\vert_A \;\colon\; B \longrightarrow Y $$ are continuous. =-- +-- {: .proof} ###### Proof Clearly the underlying diagram of underlying [[sets]] is a pushout in [[Set]]. Therefore by prop. \ref{DescriptionOfLimitsAndColimitsInTop} we need to show that the [[topological space|topology]] on $X$ is the [[final topology]] induced by the set of functions $\{i_A, i_B\}$, hence that a [[subset]] $S \subset X$ is an [[open subset]] precisely if the [[pre-images]] (restrictions) $$ i_A^{-1}(S) = S \cap A \phantom{AAA} \text{and} \phantom{AAA} i_B^{-1}(S) = S \cap B $$ are open subsets of $A$ and $B$, respectively. Now by definition of the [[subspace topology]], if $S \subset X$ is open, then the intersections $A \cap S \subset A$ and $B \cap S \subset B$ are open in these subspaces. Conversely, assume that $A \cap S \subset A$ and $B \cap S \subset B$ are open. We need to show that then $S \subset X$ is open. Consider now first the case that $A;B \subset X$ are both open open. Then by the nature of the [[subspace topology]], that $A \cap S$ is open in $A$ means that there is an open subset $S_A \subset X$ such that $A \cap S = A \cap S_A$. Since the intersection of two open subsets is open, this implies that $A \cap S_A$ and hence $A \cap S$ is open. Similarly $B \cap S$. Therefore $$ \begin{aligned} S & = S \cap X \\ & = S \cap (A \cup B) \\ & = (S \cap A) \cup (S \cap B) \end{aligned} $$ is the union of two open subsets and therefore open. Now consider the case that $A,B \subset X$ are both closed subsets. Again by the nature of the subspace topology, that $A \cap S \subset A$ and $B \cap S \subset B$ are open means that there exist open subsets $S_A, S_B \subset X$ such that $A \cap S = A \cap S_A$ and $B \cap S = B \cap S_B$. Since $A,B \subset X$ are closed by assumption, this means that $A \setminus S, B \setminus S \subset X$ are still closed, hence that $X \setminus (A \setminus S), X \setminus (B \setminus S) \subset X$ are open. Now observe that (by [[de Morgan duality]]) $$ \begin{aligned} S & = X \setminus (X \setminus S) \\ & = X \setminus ( (A \cup B) \setminus S ) \\ & = X \setminus ( (A \setminus S) \cup (B \setminus S) ) \\ & = (X \setminus (A \setminus S)) \cap (X \setminus (B \setminus S)) \,. \end{aligned} $$ This exhibits $S$ as the intersection of two open subsets, hence as open. =-- +-- {: .num_example #attach} ###### Example If $X, Y, Z$ are [[normal topological spaces]] and $h: X \to Z$ is a [[closed embedding of topological spaces]] and $f: X \to Y$ is a [[continuous function]], then in the [[pushout]] diagram in $Top$ (example \ref{PushoutInTop}) $$\array{ X & \stackrel{h}{\to} & Z \\ \mathllap{f} \downarrow & & \downarrow \mathrlap{g} \\ Y & \underset{k}{\to} & W, }$$ the space $W$ is normal and $k: Y \to W$ is a closed embedding. =-- For **proof** of this and related statements see at _[[colimits of normal spaces]]_. ### Relation with $Set$ Write [[Set]] for the [[category]] of [[sets]]. +-- {: .num_defn #ForgetfulFunctorFromTopToSet} ###### Definition Write $$ U \colon Top \longrightarrow Set $$ for the [[forgetful functor]] that sends a topological space $X = (S,\tau)$ to its underlying set $U(X) = S \in Set$ and which regards a [[continuous function]] as a plain [[function]] on the underlying sets. =-- Prop. \ref{DescriptionOfLimitsAndColimitsInTop} means in particular that: +-- {: .num_prop } ###### Proposition The category [[Top]] has all small [[limits]] and [[colimits]]. The [[forgetful functor]] $U \colon Top \to Set$ from def. \ref{ForgetfulFunctorFromTopToSet} [[preserved limit|preserves]] and [[lifted limit|lifts]] limits and colimits. =-- (But it does not [[created limit|create]] or [[reflected limit|reflect]] them.) +-- {: .num_prop} ###### Proposition The [[forgetful functor]] $U$ from def. \ref{ForgetfulFunctorFromTopToSet} has a [[left adjoint]] $disc$, given by sending a [[set]] $S$ to the corresponding [[discrete topological space]], example \ref{DiscreteTopologicalSpaceAsCoproduct} $$ Top \stackrel{\overset{disc}{\longleftarrow}}{\underset{U}{\longrightarrow}} Set \,. $$ =-- +-- {: .num_prop #topcat} ###### Proposition The [[forgetful functor]] $U$ from def. \ref{ForgetfulFunctorFromTopToSet} exhibits $Top$ as * a [[concrete category]] * a [[topological concrete category]]. =-- ### Mono-/Epimorphisms {#MonoEpiMorphisms} +-- {: .num_prop #SubspaceInclusionsAreRegularMonos} ###### Proposition **([[regular monomorphisms]] of [[topological spaces]])** In the [[category]] [[Top]] of [[topological spaces]], 1. the [[monomorphisms]] are those [[continuous functions]] which are [[injective functions]]; 1. the [[regular monomorphisms]] are the [[topological embeddings]] (i.e. those continuous functions which are [[homeomorphisms]] onto their [[images]] equipped with the [[subspace topology]]). =-- +-- {: .proof} ###### Proof Regarding the first statement: An injective continuous function $f \colon X \to Y$ clearly has the cancellation property that defines monomorphisms: for parallel continuous functions $g_1,g_2 \colon Z \to X$, if $f \circ g_1 = f \circ g_2$, then $g_1 = g_2$, because continuous functions are equal precisely if their underlying functions of sets are equal. Conversely, if $f$ has the cancellation property, then testing on points $g_1, g_2 \colon \ast \to X$ gives that $f$ is injective. Regarding the second statement: from the construction of [[equalizers]] in [[Top]] (example \ref{EqualizerInTop}) we have that these are topological subspace inclusions. Conversely, let $i \colon X \to Y$ be a [[topological subspace embedding]]. We need to show that this is the equalizer of some pair of parallel morphisms. To that end, form the [[cokernel pair]] $(i_1, i_2)$ by taking the [[pushout]] of $i$ against itself (in the category of sets, and using the [[quotient topology]] on a [[disjoint union space]]). By [this prop.](regular+monomorphism#RegEquEff), the equalizer of that pair is the set-theoretic equalizer of that pair of functions endowed with the [[subspace topology]]. Since monomorphisms in [[Set]] are regular, we get the function $i$ back, and again by example \ref{EqualizerInTop}, it gets equipped with the subspace topology. This completes the proof. =-- ### Intersections and quotients +-- {: .num_lemma #pushout} ###### Lemma The [[pushout]] in [[Top]] of any (closed/open) [[topological subspace]] inclusion $i \colon A \hookrightarrow B$, example \ref{TopologicalSubspace}, along any [[continuous function]] $f \colon A \to C$ is itself an a (closed/open) subspace $j \colon C \hookrightarrow D$. =-- For proof see [there](subspace+topology#pushout). ### Closed monoidal structure It is well known that [[Top]] is not [[cartesian closed]] (see for example at [[convenient category of topological spaces]]). It is however [[closed monoidal category|closed monoidal]]. * The [[tensor product]] $X\otimes Y$ is given by the cartesian product of the underlying spaces, equipped with the _topology of separate continuity_, formed by the sets $U\subseteq X\times Y$ such that $$ U_x \;\coloneqq\; \{y\in Y : (x,y\in U)\} $$ is an open subset of $Y$ and $$ U_x \;\coloneqq\; \{x\in X : (x,y)\in U\} $$ is an open subset of $X$. Equivalently, it is the topology such that for all spaces $Z$, a function $f:X\otimes Y\to Z$ is continuous if and only if: for every $x\in X$ the function $y\mapsto f(x,y)$ is continuous, and for every $y\in Y$, the functions $x\mapsto f(x,y)$ is continuous. * The [[internal hom]] $[X,Y]$ is given by the set of [[continuous functions]] $X\to Y$, together with the topology of pointwise convergence, generated by the (sub-basic) sets $$ S(x,V) \;\coloneqq\; \{f:X\to Y : f(x)\in V\} $$ for each $x\in X$ and each open $V\subseteq Y$. Equivalently, a net $(f_\alpha:X\to Y)$ tends to $f:X\to Y$ if and only if for all $x\in X$, $f_\alpha(x)\to f(x)$ in $Y$. ## Related concepts * [[topological concrete category]] * [[Ho(Top)]], [[∞Grpd]] * [[convenient category of topological spaces]] * [[TopGrp]] ## References For general references see those listed at _[[topology]]_, such as * {#Bourbaki71} [[Nicolas Bourbaki]], chapter 1 _Topological Structures_ of _Elements of Mathematics III: General topology_, Springer 1971, 1990 See also * {#AguilarGitlerPrieto02} Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, section 12 of _Algebraic topology from a homotopical viewpoint_, Springer (2002) ([toc pdf](http://tocs.ulb.tu-darmstadt.de/106999419.pdf)) An axiomatic desciption of $Top$ along the lines of [[ETCS]] for [[Set]] is discussed in * Dana Schlomiuk, _An elementary theory of the category of topological space_, Transactions of the AMS, volume 149 (1970) For its [closed monoidal structure](#closed_monoidal_structure), see: * Maria Cristina Pedicchio and Fabio Rossi, _Monoidal closed structures for topological spaces: counter-example to a question of Booth and Tillotson_, Cahiers de topologie et géométrie différentielle catégoriques, 24(4), 1983. * {#dagger_martingales} Appendix A of [[Paolo Perrone]] and Ruben Van Belle, _Convergence of martingales via enriched dagger categories_, 2024. 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