View source for Compactly continuous map

{{TEX|done}}{{MSC|54C10}}

A map $f$ of [[topological space]]s $X \rightarrow Y$ with the property that the restriction of $f$ to any compact subspace of $X$ is continuous.  Clearly any continuous map is compactly continuous, and the converse holds if $X$ is a [[locally compact space]].  The composite of compactly continuous maps is again compactly continuous.

The [[category]] of [[Hausdorff space]]s and compactly continuous maps is equivalent to the category of [[compactly generated space]]s and continuous maps.  It is a [[Cartesian-closed category]].

See: [[Exponential law (in topology)]] and [[Space of mappings, topological]].

==References==
* Francis Borceux, "Handbook of Categorical Algebra: Volume 2, Categories and Structures", Encyclopedia of Mathematics and its Applications, Cambridge University Press (1994) {{ISBN|0-521-44179-X}} {{ZBL|1143.18002}}
* R. Brown,   "Function spaces and product topologies"  ''Quart. J. Math.''  '''2'''  (1964)  pp. 238–250. {{ZBL|0126.38503}}