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\section*{compactly generated topological space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{kaonization}{Kaonization}\dotfill \pageref*{kaonization} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{cartesian_closure}{Cartesian closure}\dotfill \pageref*{cartesian_closure} \linebreak
\noindent\hyperlink{local_cartesian_closure}{Local cartesian closure}\dotfill \pageref*{local_cartesian_closure} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A [[topological space]] is \emph{compactly generated} if (in a certain sense) the continuous images in it of all [[compact Hausdorff space]]s tell you everything about its topology.

Compactly generated spaces form a [[convenient category of topological spaces]].

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

A [[function]] $f\colon X \to Y$ between [[topological space]]s is \textbf{$k$-continuous} if for all [[compact Hausdorff space]]s $C$ and [[continuous functions]] $t\colon C \to X$ the composite $f \circ t\colon C \to Y$ is continuous.

The following conditions on a space $X$ are equivalent:

\begin{enumerate}%
\item For all spaces $Y$ and all functions $f\colon X \to Y$, $f$ is continuous if and only if $f$ is $k$-continuous.
\item There is a [[set]] $S$ of compact Hausdorff spaces such that the previous condition holds for all $C \in S$.
\item $X$ is an [[identification space]] of a [[disjoint union]] of compact Hausdorff spaces.
\item A [[subspace]] $U \subseteq X$ is [[open subspace|open]] if and only if the [[preimage]] $t^{-1}(U)$ is open for any compact Hausdorff space $C$ and continuous $t\colon C \to X$.

\end{enumerate}
A space $X$ is a \textbf{$k$-space} if any (hence all) of the above conditions hold. Some authors also say that a $k$-space is \textbf{compactly generated}, while others reserve that term for a $k$-space which is also \emph{[[weak Hausdorff space|weak Hausdorff]]}, meaning that the image of any $t\colon C\to X$ is closed (when $C$ is compact Hausdorff). Some authors go on to require a [[Hausdorff space|Hausdorff]] space, but this seems to be unnecessary.

Sometimes $k$-spaces are called \textbf{Kelley spaces}, after [[John Kelley]], who studied them extensively; however, they predate him and the `$k$' does not stand for his name. (Probably it has something to do with `compact' or `kompakt'.)

\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

Examples of compactly generated spaces include

\begin{example}
\label{}\hypertarget{}{}
Every [[compact space]] is compactly generated.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
Every [[locally compact space]] is compactly generated.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
Every [[topological manifold]] is compactly generated

\end{example}
\begin{example}
\label{CWComplexIsCompactlyGenerated}\hypertarget{CWComplexIsCompactlyGenerated}{}
Every [[CW-complex]] is a compactly generated topological space.

\end{example}
\begin{proof}
Since a CW-complex $X$ is a [[colimit]] in [[Top]] over attachments of standard [[n-disks]] $D^{n_i}$ (its cells), by the characterization of colimits in $Top$ (\href{Top#DescriptionOfLimitsAndColimitsInTop}{prop.}) a subset of $X$ is open or closed precisely if its restriction to each cell is open or closed, respectively. Since the $n$-disks are compact, this implies one direction: if a subset $A$ of $X$ intersected with all compact subsets is closed, then $A$ is closed.

For the converse direction, since [[a CW-complex is a Hausdorff space]] and since [[compact subspaces of Hausdorff spaces are closed]], the intersection of a closed subset with a compact subset is closed.

\end{proof}
\begin{example}
\label{}\hypertarget{}{}
Every [[first countable space]] is a compactly generated space.

\end{example}
\begin{proof}
Since the topology is determined by convergent sequences = maps from one-point [[compactification]] $\mathbb{N} \cup \{\infty\}$); these include all Fr\'e{}chet spaces.

\end{proof}
\hypertarget{kaonization}{}\subsection*{{Kaonization}}\label{kaonization}

Let $k\Top$ denote the category of $k$-spaces and continuous maps, and $\Top_k$ denote the category of all topological spaces and $k$-continuous maps. We have inclusions

\begin{displaymath}
k\Top \to \Top \to \Top_k
\end{displaymath}
of which the first is the inclusion of a [[full subcategory|full]] [[coreflective subcategory]], the second is [[bijective on objects functor|bijective on objects]], and the composite $k\Top \to Top_k$ is an [[equivalence of categories]].

The [[coreflection]] $\Top \to k\Top$ is denoted $k$, and is sometimes (e.g. by [[M M Postnikov]]) also called \textbf{kaonization} and sometimes (e.g. by [[Peter May]]) \textbf{$k$-ification}. This functor is constructed as follows: we take $k(X)=X$ as a set, but with the topology whose closed sets are those whose intersection with compact Hausdorff subsets of (the original topology on) $X$ is closed (in the original topology on $X$). Then $k(X)$ has all the same closed sets and possibly more, hence all the same open sets and possibly more.

In particular, the identity map $id:k(X)\to X$ is continuous, and forms the counit of the coreflection. Thus this coreflection has a counit which is both [[monic]] and [[epic]], i.e. a ``[[bimorphism]]''---such a coreflection is sometimes called a ``bicoreflection.''

Moreover, the identity $id: X \to k(X)$ is $k$-continuous, so that the counit becomes an isomorphism in $\Top_k$. This shows that $k\Top \to \Top_k$ is [[essentially surjective functor|essentially surjective]], and it is fully faithful since any $k$-continuous function between $k$-spaces is $k$-continuous; hence it is an equivalence.

Since $k\Top \hookrightarrow \Top$ is coreflective, it follows that $k\Top$ is [[complete category|complete]] and [[cocomplete category|cocomplete]]. Its [[colimits]] are constructed as in $Top$, but its [[limits]] are the $k$-ification of limits in $Top$. This is nontrivial already for [[products]]: the $k$-space product $X\times Y$ is the $k$-ification of the usual [[product topology]]. The $k$-space product is better behaved in many ways; e.g. it enables [[geometric realization]] to preserve products (and all finite limits), and the product of two [[CW complexes]] to be another CW complex.

If one is interested in $k$-spaces which are also [[weakly Hausdorff space|weak Hausdorff]], then there is a further [[reflector]] which must be applied; see [[weakly Hausdorff space]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{cartesian_closure}{}\subsubsection*{{Cartesian closure}}\label{cartesian_closure}

The categories $k\Top\simeq \Top_k$ are [[cartesian closed category|cartesian closed]]. (While in [[Top]] only some objects are exponentiable, see [[exponential law for spaces]].) For arbitrary spaces $X$ and $Y$, define the \emph{test-open} or \emph{compact-open topology} on $\Top_k(X,Y)$ to have the [[subbase]] of sets $M(t,U)$, for a given compact Hausdorff space $C$, a map $t\colon C \to X$, and an open set $U$ in $Y$, where $M(t,U)$ consists of all $k$-continuous functions $f\colon X \to Y$ such that $f(t(C))\subseteq U$.

(This is slightly different from the usual [[compact-open topology]] if $X$ happens to have non-Hausdorff compact subspaces; in that case the usual definition includes such subspaces as tests, while the above definition excludes them. Of course, if $X$ itself is Hausdorff, then the two become identical.)

With this topology, $\Top_k(X,Y)$ becomes an [[exponential object]] in $Top_k$. It follows, by [[Yoneda lemma]] arguments (\href{closed+monoidal+category#TensorHomIsoInternalizes}{prop.}), that the bijection

\begin{displaymath}
k\Top(X \times Y, Z) \to kTop(X,k\Top(Y,Z))
\end{displaymath}
is actually an isomorphism in $\Top_k$, which we may call a \emph{$k$-homeomorphism} (e.g. \hyperlink{Strickland09}{Strickland 09, prop. 2.12}). In fact, it is actually a homeomorphism, i.e. an isomorphism already in $Top$.

[[Zoran Škoda]]: I do not understand the remark. I mean if the domain is k-space then by the characterization above continuous is the same as k-continuous. Thus if both domain and codomain are continuous then homeo is the same as k-homeo. I assume that even in noHausdorff case, the test-open topology for $X$ and $Y$ k-spaces gives a k-space and that the cartesian product has the correction for the k-spaces.

[[Todd Trimble]]: That may be just the point: that the domain is not necessarily a $k$-space. I have to admit that I haven't worked through the details of this exposition, but one thing I tripped over is the fact that we're dealing with \emph{all} topological spaces $X$, $Y$, not just $k$-spaces.

[[Mike Shulman]]: But any topological space is isomorphic in $k\Top$ to its $k$-ification, right? So $k\Top$ might as well be defined to consist of $k$-spaces and continuous maps.

[[Todd Trimble]]: Okay, you're right that makes sense. So in that case, it seems that Zoran definitely has a point here.

[[Mike Shulman]]: See the \href{https://nforum.ncatlab.org/comments.php?DiscussionID=1958}{nForum discussion}.

It follows that the category $k\Top$ of $k$-spaces and continuous maps is also cartesian closed, since it is equivalent to $\Top_k$. Its exponential object is the $k$-ification of the one constructed above for $\Top_k$. Since for $k$-spaces, $k$-continuous implies continuous, the underlying set of this exponential space $k\Top(X,Y)$ is the set of all continuous maps from $X$ to $Y$. Thus, when $X$ is Hausdorff, we can identify this space with the $k$-ification of the usual [[compact-open topology]] on $Top(X,Y)$.

Finally, this all remains true if we also impose the weak Hausdorff, or Hausdorff, conditions.

\hypertarget{local_cartesian_closure}{}\subsubsection*{{Local cartesian closure}}\label{local_cartesian_closure}

Unfortunately neither of the above categories is [[locally cartesian closed category|locally cartesian closed]].

However, if $K$ is the category of not-necessarily-weak-Hausdorff k-spaces, and $A$ and $B$ are k-spaces that are weak Hausdorff, then the pullback functor $K/B\to K/A$ has a right adjoint. This is what May and Sigurdsson used in their book \emph{Parametrized homotopy theory}.

There is still a lot of work on fibred exponential laws and their applications. One reason for the success and difficulties is that it is easy to give a topology on the space of closed subsets of a space $X$ by regarding this as the space of maps to the [[Sierpinski space]] (the set $\{0,1\}$ of [[truth value]]s in which $\{1\}$ is closed but not open). From this one can get an [[exponential law for spaces]] over $B$ if $B$ is $T_0$, so that all fibres of spaces over $B$ are closed in their total space. Note that weak Hausdorff implies $T_0$.

[[Mike Shulman]]: What precisely does ``get an exponential law'' mean? Do you mean that $k Top/B$ is cartesian closed if $B$ is $T_0$?

\emph{Toby}: Hopefully that is explained in the new article.

\emph{Mike}: Which new article? [[exponential law for spaces]]? That page doesn't talk about fibered exponentials at all.

\emph{Toby}: Seeing this later, I no longer know what article I meant.

\hypertarget{references}{}\subsection*{{References}}\label{references}

The following article attributes the concept to Hurewicz:

\begin{itemize}%
\item David Gale, \emph{Compact Sets of Functions and Function Rings} , Proc. AMS \textbf{1} (1950) pp.303-308. (\href{http://www.ams.org/journals/proc/1950-001-03/S0002-9939-1950-0036503-X/S0002-9939-1950-0036503-X.pdf}{pdf})

\end{itemize}
Compactly generated spaces are discussed by J. L. Kelley in his book

\begin{itemize}%
\item [[John Kelley]], \emph{General topology}, D. van Nostrand, New York 1955.

\end{itemize}
An early textbook account is in

\begin{itemize}%
\item [[Pierre Gabriel]], [[Michel Zisman]], sections I.1.5.3 and III.2 of \emph{[[Calculus of fractions and homotopy theory]]}, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967) (\href{https://www.math.rochester.edu/people/faculty/doug/otherpapers/GZ.pdf}{pdf})

\end{itemize}
A lectue note careful about the (weakly) Hausdorff assumptions when needed/wanted is in the lecture notes

\begin{itemize}%
\item [[Neil Strickland]], \emph{The category of CGWH spaces}, 2009 (\href{http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf}{pdf})

\end{itemize}
Many properties of compactly generated Hausdorff spaces are used to establish a variant of the theory of fibrations, cofibrations and deformation retracts in

\begin{itemize}%
\item [[Norman Steenrod]], \emph{A convenient category of topological spaces}, Michigan Math. J. 14 (1967) 133--152, \href{http://projecteuclid.org/euclid.mmj/1028999711}{project euclid}

\end{itemize}
Gabriel and Zisman discuss the connection with the exactness of [[geometric realization]] in

\begin{itemize}%
\item [[Peter Gabriel]], Michel Zisman, \emph{Calculus of Fractions and Homotopy Theory} , Springer Heidelberg 1967. (ch.III.3-4)

\end{itemize}
Other and later references include

\begin{itemize}%
\item [[Gaunce Lewis]], \emph{Compactly generated spaces} (\href{http://www.math.uchicago.edu/~may/MISC/GaunceApp.pdf}{pdf}), appendix A of \emph{The Stable Category and Generalized Thom Spectra} PhD thesis Chicago, 1978


\item [[George Whitehead]], \emph{Elements of homotopy theory}


\item [[Brian Day|Brian J. Day]], \emph{Relationship of Spanier's Quasi-topological Spaces to k-Spaces} , M. Sc. thesis University of Sydney 1968. (\href{http://www.math.mq.edu.au/~street/DayMasters.pdf}{pdf})


\item Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, around note 4.3.22 of \emph{Algebraic topology from a homotopical viewpoint}, Springer (2002) (\href{http://tocs.ulb.tu-darmstadt.de/106999419.pdf}{toc pdf})


\item [[Ronnie Brown]], \emph{Topology and groupoids}, Booksurge 2006, section 5.9.


\item Booth, Peter I.; Heath, Philip R.; Piccinini, Renzo A. Fibre preserving maps and functional spaces. Algebraic topology (Proc. Conf., Univ. British Columbia, Vancouver, B.C., 1977), pp. 158--167, Lecture Notes in Math., 673, Springer, Berlin, 1978.


\item [[Peter May]], \emph{[[A concise course in algebraic topology]]}, Chapter 5


\item Samuel Smith, \emph{The homotopy theory of function spaces: a survey} (\href{http://arxiv.org/abs/1009.0804}{arXiv:1009.0804})


\item [[Stefan Schwede]], section A.2 of \emph{[[Symmetric spectra]]} (2012)



\end{itemize}
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