# Compactly continuous map

From Encyclopedia of Mathematics

2010 Mathematics Subject Classification: *Primary:* 54C10 [MSN][ZBL]

A map $f$ of topological spaces $X \rightarrow Y$ 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 spaces and compactly continuous maps is equivalent to the category of compactly generated spaces 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

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Compactly continuous map.

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