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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{compact-open topology} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{topology}{}\paragraph*{{Topology}}\label{topology} [[!include topology - contents]] \hypertarget{mapping_space}{}\paragraph*{{Mapping space}}\label{mapping_space} [[!include mapping space - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{Exponentiability}{Exponentiability}\dotfill \pageref*{Exponentiability} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{maps_out_of_the_point_space}{Maps out of the point space}\dotfill \pageref*{maps_out_of_the_point_space} \linebreak \noindent\hyperlink{maps_into_metric_spaces}{Maps into metric spaces}\dotfill \pageref*{maps_into_metric_spaces} \linebreak \noindent\hyperlink{loop_spaces_and_path_spaces}{Loop spaces and path spaces}\dotfill \pageref*{loop_spaces_and_path_spaces} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A natural [[topology]] on [[mapping spaces]] of [[continuous functions]], important because of its role in exhibiting [[locally compact topological spaces]] to be [[exponential object|exponentiable]], as demonstrated below, culminating in Corollary \ref{GivesExponentialObject}. The compact-open topology on the set of [[continuous functions]] $X \to Y$ is generated by the [[base of a topology|subbasis]] of subsets $U^K \subset C(X,Y)$ that map a given [[compact subspace]] $K \subset X$ to a given [[open subset]] $U \subset Y$, whence the name. When restricting to continuous functions between [[compactly generated topological spaces]] one usually modifies this definition to a subbase of open subsets $U^{\phi(K)}$, where now $\phi(K)$ is the [[image]] of a [[compact topological space]] under any continuous function $\phi \colon K \to X$. This definition gives a [[cartesian closed category|cartesian]] [[internal hom]] in the category of [[compactly generated topological spaces]] (see also at \emph{[[convenient category of topological spaces]]}). The two definitions agree when the [[domain]] $X$ is a [[compactly generated topological space|compactly generated]] [[Hausdorff space]], but not in general. Beware that texts on compactly generated spaces nevertheless commonly say ``compact-open topology'' for the second definition. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \begin{note} \label{}\hypertarget{}{} Let $(X, \mathcal{O}_{X})$ and $(Y, \mathcal{O}_{Y})$ be [[topological spaces]]. We denote by $Y^{X}$ the set of [[continuous maps]] from $(X,\mathcal{O}_{X})$ to $(Y, \mathcal{O}_{Y})$. \end{note} \begin{remark} \label{}\hypertarget{}{} Other common notations for $Y^{X}$ are $Map(X,Y)$ or $C(X,Y)$. \end{remark} \begin{note} \label{}\hypertarget{}{} Let $(X, \mathcal{O}_{X})$ be a [[topological space]]. We denote by $\mathcal{O}^{c}_{X}$ the set of subsets of $X$ which are [[compact subset|compact]] with respect to $\mathcal{O}_{X}$. \end{note} \begin{note} \label{}\hypertarget{}{} Let $(X, \mathcal{O}_{X})$ be a [[topological space]]. Let $A$ be a subset of $X$. We denote by $\overline{A}$ the [[topological closure]] of $A$ in $X$ with respect to $\mathcal{O}_{X}$. \end{note} \begin{defn} \label{LocallyCompact}\hypertarget{LocallyCompact}{} A topological space $(X, \mathcal{O}_{X})$ is \emph{[[locally compact topological space|locally compact]]} if the set of compact neighbourhoods of points of $X$ form a [[neighbourhood basis]] of $X$. That is to say: for every $x \in X$ and every [[neighbourhood]] $N$ of $x$, there is a $V \in \mathcal{O}^{c}_{X}$ and a $U \in \mathcal{O}_{X}$ such that $x \in U \subset V \subset N$. \end{defn} \begin{remark} \label{}\hypertarget{}{} There are many variations on this definition, which can be found at \emph{[[locally compact space]]}. These are all equivalent if $(X, \mathcal{O}_{X})$ is [[Hausdorff space|Hausdorff]]. We do \emph{not} however make the assumption that $(X, \mathcal{O}_{X})$ is [[Hausdorff space|Hausdorff]]. \end{remark} \begin{note} \label{}\hypertarget{}{} Let $(X, \mathcal{O}_{X})$ and $(Y, \mathcal{O}_{Y})$ be [[topological spaces]]. Given $A \in \mathcal{O}^{c}_{X}$ and $U \in \mathcal{O}_{Y}$, we denote by $M_{A,U}$ the set of continuous maps $f : X \rightarrow Y$ such that $f(A) \subset U$. \end{note} \begin{defn} \label{CompactOpenTopology}\hypertarget{CompactOpenTopology}{} Let $(X, \mathcal{O}_{X})$ and $(Y, \mathcal{O}_{Y})$ be [[topological spaces]]. The \emph{compact-open} topology on $Y^{X}$ is that with [[base of a topological space|sub-basis]] given by the set of sets $M_{A,U}$ such that $A \in \mathcal{O}^{c}_{X}$ and $U \in \mathcal{O}_{Y}$. \end{defn} \begin{note} \label{CompactOpenTopologyNotation}\hypertarget{CompactOpenTopologyNotation}{} Let $(X, \mathcal{O}_{X})$ and $(Y, \mathcal{O}_{Y})$ be [[topological spaces]]. We shall denote the compact-open topology (def. \ref{CompactOpenTopology}) on $Y^{X}$ by $\mathcal{O}_{Y^{X}}$. \end{note} \hypertarget{Exponentiability}{}\subsection*{{Exponentiability}}\label{Exponentiability} \begin{prop} \label{}\hypertarget{}{} Let $(X, \mathcal{O}_{X})$ be a locally compact topological space, def. \ref{LocallyCompact}, and let $(Y, \mathcal{O}_{Y})$ be a [[topological space]]. The map $ev : X \times Y^{X} \rightarrow Y$ given by $(x,f) \mapsto f(x)$ is continuous, where $X \times Y^{X}$ is equipped with the [[product topology]] $\mathcal{O}_{X \times Y^{X}}$ with respect to $\mathcal{O}_{X}$ and $\mathcal{O}_{Y^{X}}$ (notation \ref{CompactOpenTopologyNotation}). \end{prop} \begin{proof} By an elementary fact concerning continuous maps, it suffices to show that for any $(x,f) \in X \times Y^{X}$, and any $U \in \mathcal{O}_{Y}$ such that $f(x) \in U$, there is a $U' \in \mathcal{O}_{X \times Y^{X}}$ such that $(x,f) \in U'$, and such that $ev(U') \subset U$. To demonstrate this, we make the following observations. 1) Since $f$ is continuous, we have that $f^{-1}(U) \in \mathcal{O}_{X}$. 2) Since $(X, \mathcal{O}_{X})$ is locally compact, we deduce from 1) that there is a $V \in \mathcal{O}^{c}_{X}$ and a $U'' \in \mathcal{O}_{X}$ such that $x \in U'' \subset V \subset f^{-1}(U)$. 3) By 2) and by definition of $\mathcal{O}_{X \times Y^{X}}$ and $\mathcal{O}_{Y^{X}}$, we have that $U'' \times M_{V, U} \in \mathcal{O}_{X \times Y^{X}}$, and that $(x,f) \in U'' \times M_{V, U}$. 4) Let $(x',f') \in U'' \times M_{V, U}$, Since $x' \in U''$, and since $f'(U'') \subset f'(V) \subset U$, the latter inclusion holding by definition of $M_{V, U}$, we have that $f'(x) \in U$. We deduce that $ev(U'' \times M_{ V, U}) \subset U$. By 3) and 4), we see that we can take $U'$ as the beginning of the proof to be $U'' \times M_{V , U}$. \end{proof} \begin{prop} \label{alphafIsContinuous}\hypertarget{alphafIsContinuous}{} Let $(X, \mathcal{O}_{X})$, $(Y, \mathcal{O}_{Y})$, and $(Z, \mathcal{O}_{Z})$ be [[topological spaces]]. Let $f : X \times Y \rightarrow Z$ be a continuous map, where $X \times Y$ is equipped with the [[product topology]] $\mathcal{O}_{X \times Y}$ with respect to $\mathcal{O}_{X}$ and $\mathcal{O}_{Y}$. Then the map $\alpha_{f} : X \rightarrow Z^{Y}$ given by $x \mapsto f_{x}$ is continuous, where $f_{x} : Y \rightarrow Z$ is given by $y \mapsto f(x,y)$, and where $Z^{Y}$ is equipped with the compact-open topology $\mathcal{O}_{Z^{Y}}$. \end{prop} \begin{proof} By an elementary fact concerning continuous maps, it suffices to show that for any $x \in X$, and any $M_{A,U} \in \mathcal{O}_{Z^{Y}}$ such that $f_{x} \in M_{A,U}$, there is a $U' \in \mathcal{O}_{X}$ such that $x \in U'$, and such that $\alpha_{f}( U' ) \subset M_{A,U}$. To demonstrate this, we make the following observations. 1) Since $f_{x} \in M_{A,U}$, we have, by definition of $f_{x}$ and by definition of $M_{A,U}$, that $f(x,a) \in U$ for all $a \in A$. 2) Since $f$ is continuous, we have that $f^{-1}(U) \in \mathcal{O}_{X \times Y}$. 3) By 1), we have that $\{x\} \times A \subset f^{-1}(U)$. 4) Let $\mathcal{O}_{X \times A}$ denote the [[subspace topology]] on $X \times A$ with respect to $\mathcal{O}_{X \times Y}$. By 2), we have that $f^{-1}(U) \cap (X \times A) \in \mathcal{O}_{X \times A}$. 5) By 3), we have that $\{x \} \times A \subset f^{-1}(U) \cap (X \times A)$. 6) Since $A \in \mathcal{O}^{c}_{Y}$, it follows from 4), 5), and the [[tube lemma]] that there is a $U' \in \mathcal{O}_{X}$ such that $x \in U'$ and such that $U' \times A \subset f^{-1}(U) \cap (X \times A) \subset f^{-1}(U)$. 7) We deduce from 6) that $f(U' \times A) \subset U$. This is the same as to say that $f_{x'}(a) \in U$ for all $x' \in U'$. Thus $f_{x'}$ belongs to $M_{A,U}$ for all $x' \in U'$, which is the same as to say that $\alpha_{f}(U') \subset M_{A,U}$. We conclude that we can take the required $U'$ of the beginning of the proof to be the $U'$ of 6). \end{proof} \begin{prop} \label{alphainvfIsContinuous}\hypertarget{alphainvfIsContinuous}{} Let $(X, \mathcal{O}_{X})$, $(Y, \mathcal{O}_{Y})$, and $(Z, \mathcal{O}_{Z})$ be [[topological spaces]]. Suppose that $(Y, \mathcal{O}_{Y})$ is locally compact. Let $f : X \rightarrow Z^{Y}$ be a continuous map, where $Z^{Y}$ is equipped with the compact-open topology $\mathcal{O}_{Z^{Y}}$. Then the map $\alpha^{-1}_{f} : X \times Y \rightarrow Z$ given by $(x,y) \mapsto \big( f(x) \big)(y)$ is continuous, where $X \times Y$ is equipped with the [[product topology]] $\mathcal{O}_{X \times Y}$. \end{prop} \begin{proof} We make the following observations. 1) We have that $\alpha^{-1}_{f} = ev \circ \tau \circ (f \times id)$, where $\tau : Z^{Y} \times Y \rightarrow Y \times Z^{Y}$ is given by $(f,y) \mapsto (y,f)$. 2) By Proposition \ref{alphafIsContinuous}, we have that $ev$ is continuous. 3) It is an elementary fact that $\tau$ is continuous. 4) Since $f$ is continuous, it follows by an elementary fact that $f \times id$ is continuous. 5) We deduce from 2) - 4) that $ev \circ \tau \circ (f \times id)$ is continuous. By 1), we conclude that $\alpha^{-1}_{f}$ is continuous. \end{proof} \begin{cor} \label{GivesExponentialObject}\hypertarget{GivesExponentialObject}{} Let $(X, \mathcal{O}_{X})$ and $(Y, \mathcal{O}_{Y})$ be [[topological spaces]]. Suppose that $(X, \mathcal{O}_{X})$ is locally compact. Then $(Y^{X}, \mathcal{O}_{Y^{X}})$ together with the corresponding map $ev$ defines an [[exponential object]] in the [[category]] [[Top]] of all topological spaces. \end{cor} \begin{proof} Follows immediately from Proposition \ref{alphafIsContinuous}, Proposition \ref{alphainvfIsContinuous}, and the fact that $Y^{X}$ and $ev$ are exhibited by the corresponding maps $\alpha_{-}$ and $\alpha^{-1}_{-}$ of Proposition \ref{alphafIsContinuous} and Proposition \ref{alphainvfIsContinuous} to define an exponential object in the category $\mathsf{Set}$ of sets. \end{proof} \begin{remark} \label{}\hypertarget{}{} A proof can also be found in \hyperlink{AguilarGitlerPrieto02}{Aguilar-Gitler-Prieto 02, prop. 1.3.1}, or just about any half-decent textbook on point-set topology! However, the result is almost universally stated with an assumption that $(X,\mathcal{O}_{X})$ is [[Hausdorff space|Hausdorff]] which, as the proof we have given illustrates, is not needed. \end{remark} \begin{remark} \label{}\hypertarget{}{} We moreover have a homeomorphism $Map(X,Map(Y,Z))\cong Map(X\times Y,Z)$ if in addition $X$ is [[Hausdorff space|Hausdorff]]. See also [[convenient category of topological spaces]]. \end{remark} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{maps_out_of_the_point_space}{}\subsubsection*{{Maps out of the point space}}\label{maps_out_of_the_point_space} \begin{example} \label{MappingSpaceOutOfPoint}\hypertarget{MappingSpaceOutOfPoint}{} \textbf{([[mapping space]] construction out of the [[point space]] is the identity)} The [[point space]] $\ast$ is clearly a [[locally compact topological space]]. Hence for every [[topological space]] $(X,\tau)$ the mapping space $Maps(\ast, (X,\tau))$ exists. This is [[homeomorphism|homeomorphic]] to the space $(x,\tau)$ itself: \begin{displaymath} Maps(\ast, (X,\tau)) \simeq (X,\tau) \,. \end{displaymath} \end{example} \hypertarget{maps_into_metric_spaces}{}\subsubsection*{{Maps into metric spaces}}\label{maps_into_metric_spaces} If $Y$ is a [[metric space]] then the compact-open topology on $Map(X,Y)$ is the \emph{topology of uniform convergence on compact subsets} in the sense that $f_n \to f$ in $Map(X,Y)$ with the compact-open topology iff for every compact subset $K\subset X$, $f_n \to f$ uniformly on $K$. If (in addition) the domain $X$ is compact then this is the \emph{[[topology of uniform convergence]]}. \hypertarget{loop_spaces_and_path_spaces}{}\subsubsection*{{Loop spaces and path spaces}}\label{loop_spaces_and_path_spaces} \begin{example} \label{LoopSpace}\hypertarget{LoopSpace}{} \textbf{([[loop space]] and [[path space]])} Let $(X,\tau)$ be any [[topological space]]. \begin{enumerate}% \item The standard [[circle]] $S^1$ is a [[compact Hausdorff space]] [[open subspaces of compact Hausdorff spaces are locally compact|hence]] a [[locally compact topological space]]. Accordingly the [[mapping space]] \begin{displaymath} \mathcal{L} X \coloneqq Maps( S^1, (X,\tau) ) \end{displaymath} exists. This is called the \emph{[[free loop space]]} of $(X,\tau)$. If both $S^1$ and $X$ are equipped with a choice of point (``[[basepoint]]'') $s_0 \in S^1$, $x_0 \in X$, then the [[topological subspace]] \begin{displaymath} \Omega X \subset \mathcal{L}X \end{displaymath} on those functions which take the basepoint of $S^1$ to that of $X$, is called the \emph{[[loop space]]} of $X$, or sometimes \emph{[[based loop space]]}, for emphasis. \item Similarly the [[closed interval]] is a [[compact Hausdorff space]] [[open subspaces of compact Hausdorff spaces are locally compact|hence]] a [[locally compact topological space]] (def. \ref{LocallyCompactSpace}). Accordingly the [[mapping space]] \begin{displaymath} Maps( [0,1], (X,\tau) ) \end{displaymath} exists. Again if $X$ is equipped with a choice of basepoint $x_0 \in X$, then the [[topological subspace]] of those functions that take $0 \in [0,1]$ to that chosen basepoint is called the \emph{[[path space]]} of $(X\tau)$: \begin{displaymath} P X \subset Maps( [0,1], (X,\tau) ) \,. \end{displaymath} \end{enumerate} Notice that we may encode these subspaces more abstractly in terms of [[universal properties]]: The path space and the loop space are characterized, up to [[homeomorphisms]], as being the [[limit|limiting cones]] in the following [[pullback]] diagrams of topological spaces: \begin{enumerate}% \item [[loop space]]: \begin{displaymath} \itexarray{ \Omega X &\longrightarrow& Maps(S^1, (X,\tau)) \\ \downarrow &(pb)& \downarrow^{\mathrlap{Maps(const_{s_0}, id_{(X,\tau)})}} \\ \ast &\underset{const_{x_0}}{\longrightarrow}& X \simeq Maps(\ast,(X,\tau)) } \,. \end{displaymath} \item [[path space]]: \begin{displaymath} \itexarray{ P X &\longrightarrow& Maps([0,1], (X,\tau)) \\ \downarrow &(pb)& \downarrow^{\mathrlap{Maps(const_x, id_{(X,\tau)})}} \\ \ast &\underset{const_{x_0}}{\longrightarrow}& X \simeq Maps(\ast,(X,\tau)) } \end{displaymath} \end{enumerate} Here on the right we are using that the mapping space construction is a [[functor]] and we are using example \ref{MappingSpaceOutOfPoint} in the identification on the bottom right mapping space out of the point space. \end{example} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[topology of mapping spaces]] \item [[differential topology of mapping spaces]] \item [[C-k topology]] \item [[space of knots]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item Ralph H. Fox, \emph{On Topologies for Function Spaces} , Bull. AMS \textbf{51} (1945) pp.429-432. (\href{http://projecteuclid.org/download/pdf_1/euclid.bams/1183506987}{pdf}) \item Eva Lowen-Colebunders, G\"u{}nther Richter, \emph{An Elementary Approach to Exponential Spaces}, Applied Categorical Structures May 2001, Volume 9, Issue 3, pp 303-310 (\href{http://link.springer.com/article/10.1023%2FA%3A1011268007097}{publisher}) \item Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, sections 1.2, 1.3 of \emph{Algebraic topology from a homotopical viewpoint}, Springer (2002) (\href{http://tocs.ulb.tu-darmstadt.de/106999419.pdf}{toc pdf}) \item \href{http://en.wikipedia.org/wiki/Compact-open_topology}{Wikipedia entry} \end{itemize} [[!redirects compact open topology]] [[!redirects space of maps]] [[!redirects spaces of maps]] \end{document}