User:Richard Pinch/sandbox-9

Core-compact space

Let $X$ be a topological space with $\mathfrak{O}_X$ the collection of open sets. If $U, V$ are open, we say that $U$ is compact in $V$ if every open cover of $V$ has a finite subset that covers $U$. The 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 compact in $V$.

A space is core compact if and only if $\mathfrak{O}_X$ 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.