# Compactly generated space

From Encyclopedia of Mathematics

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2010 Mathematics Subject Classification: *Primary:* 54B30 *Secondary:* 18F60 [MSN][ZBL]

*Kelley space, $k$-space*

A Hausdorff topological space in which a subset is closed if its intersection with any compact subset is closed. Every locally compact Hausdorff space is compactly generated, as is every first countable Hausdorff space.

The category of compactly generated spaces and continuous maps is equivalent to the category of Hausdorff spaces and compactly 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

**How to Cite This Entry:**

Compactly generated space.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Compactly_generated_space&oldid=51464