View source for Core-compact space

{{TEX|done}}{{MSC|54D30|54D50,18F60}}

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 set|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}}