# Core-compact space

2010 Mathematics Subject Classification: *Primary:* 54D30 *Secondary:* 54D5018F60 [MSN][ZBL]

A topological space $X$ is core compact if for any $x \in X$ and open neighbourhood $N$ of $x$, there is an open set $V$ such that $N$ is relatively compact in $V$ (every open cover of $V$ has a finite subset that covers $N$); equivalently, $N$ is way below $X$.

A space is core compact if and only if the collection of open sets $\mathfrak{O}_X$ is a continuous lattice. A locally compact space is core compact, and a sober space (and hence in particular a Hausdorff space) is core compact if and only if it is locally compact.

A space is core compact if and only if the product of the identity with a quotient map is quotient. The core compact spaces are precisely the exponentiable spaces in the category of topological spaces; that is, the spaces $X$ such that ${-} \times X$ has a right adjoint ${-}^X$. See Exponential law (in topology).

## References

- Dirk Hofmann, Gavin J. Seal, Walter Thole (edd.) "Monoidal topology. A categorical approach to order, metric, and topology." Encyclopedia of Mathematics and its Applications
**153**Cambridge: Cambridge University Press (2014) (English) ISBN 978-1-107-06394-5 Zbl 1297.18001

**How to Cite This Entry:**

Core-compact space.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Core-compact_space&oldid=51472