View source for Compactly generated space

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''Kelley space, $k$-space''

A [[Hausdorff space|Hausdorff]] [[topological space]] in which a subset is closed if its intersection with any compact subset is closed.  Every [[locally compact space|locally compact]] Hausdorff space is compactly generated, as is every [[First axiom of countability|first countable]] Hausdorff space. 

The category of compactly generated spaces and continuous maps is equivalent to the category of Hausdorff spaces and [[compactly continuous map]]s.  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}}