# Difference between revisions of "Compactly generated space"

From Encyclopedia of Mathematics

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+ | ''Kelley space, $k$-space'' | ||

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+ | 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. | ||

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+ | 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]]. | ||

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

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+ | ====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}} |

## Latest revision as of 18:22, 20 January 2021

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=42389