Compactly generated space

2020 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