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Let CGWH be the category of weak Hausdorff compactly generated spaces; see e.g.

N.P. Strickland. THE CATEGORY OF CGWH SPACES: Preprint available from https://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf. 2009

or

Tammo tom Dieck. Algebraic topology. Zurich: European Mathematical Society (EMS), 2008. (7.9)

CGWH is well known to be a cartesian closed category (references above prove this). However, there appears to be two different ways to define the topology on the space TOP(X,Y) of continuous maps. Strickland topologises TOP(X,Y) with the k-ification of the topology with sub-basis $$\{f \colon X \to Y \,\, |\,\, f(u(K))\subset A\},$$ where $u\colon K \to X$ is a continuous map, where $K$ is compact Hausdorff, and $A\subset Y$ is open. In the CGWH case, and by using Lemma 1.4(b) of Strickland's paper, this coincides with the definition in May's "A concise course in Algebraic topology" chapter 5, where TOP(X,Y) is topologised with the k-ification of the topology with sub-basis: $$\{f \colon X \to Y \,\,|\,\, f(K)\subset A\},$$ where $K$ is a compact Hausdorff subset of $X$, and $A\subset Y$ is open.

However, tom Dieck topologises $TOP(X,Y)$ with the k-ification of the topology with sub-basis $$\{f \colon X \to Y \,\,|\,\, f(K)\subset A\},$$ where $K$ is a compact (not necessarily Hausdorff, it seems) subset of $X$, and $A$ is open. This is also the "official" function space topology in appendix A1 of Rudolf Fritsch and Renzo A. Piccinini. Cellular structures in topology, volume 19 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1990.

These two topologies in TOP(X,Y) are clearly the same if $X$ is Hausdorff. Do they coincide in general?

EDIT: As Ivan Yudin mentioned below, given that we have the adjunction between $(\_)\times Y$ and $C(Y,\_)$, where $X \times Y$ is topologised with the k-ification of the product topology, it then follows, assuming all references (and my interpretation of them) are correct that given CGWH spaces $X$ and $Y$ the two topologies on $TOP(X,Y)$, which k-ify the topologies with sub-basis $\{f \colon X \to Y \,\,|\,\, f(K)\subset A\}$ $A\subset Y$ open, and $K \subset X$ compact, and with sub-basis $\{f \colon X \to Y \,\,|\,\, f(K)\subset A\}$ $A\subset Y$ open, and $K \subset X$ compact and Hausdorff coincide.

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    $\begingroup$ Do both sources impose the same topology on cartesian products? AFAIU, the topology on TOP(X,Y) is uniquely determined from the adjunction between TOP(X,-) and (X × - ) and toplogies on the spaces X × Z with Z been CGWH. $\endgroup$
    – Ivan Yudin
    Jul 14, 2019 at 16:46
  • $\begingroup$ Yes. Both impose that the topology is the k-ification of the product topology. $\endgroup$ Jul 14, 2019 at 17:25
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    $\begingroup$ tom Dieck's proof is not convincing to me. The key claim is 7.9.16, which says that the evaluation map of $TOP(Y,Z):=kF(X,Y)$ in CGWH spaces is continuous. The proof relies on 2.4.3, which seems to to require as a hypothesis that the evaluation map of $F(Y,Z)$ is continuous in Top, where this is the space of cts maps topologized with subbasis that you indicated above (based on cpt but not nec Haus $K\subset Y$). It's hard to tell for sure, because the statement of 2.4.3 is confusing to me. Anyway, the argument feels circular to me. $\endgroup$ Jul 14, 2019 at 18:53
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    $\begingroup$ The following paper is relevant: msp.org/pjm/1980/88-1/p03.xhtml $\endgroup$ Jul 14, 2019 at 20:35

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