Core of a locally compact space

In topology, the core of a locally compact space is a cardinal invariant of a locally compact space $X$, denoted by $\mathrm{cor}(X)$. Locally compact spaces with countable core generalize σ-compact locally compact spaces.

The concept was introduced by Alexander Arhangel'skii.

Core of a locally compact space

Let $X$ be a locally compact and Hausdorff space. A subset $S\subseteq X$ is called saturated if it is closed in $X$ and satisfies $S\cap P\ne \mathrm{\varnothing }$ for every closed, non-compact subset $P\subseteq X$.

The core $\mathrm{cor}(X)$ is the smallest cardinal $\tau$ such that there exists a family $\gamma =({\gamma }_j)$ of saturated subsets of $X$ satisfying: $|\gamma |\le \tau$ and $\bigcap _j{\gamma }_j=\mathrm{\varnothing }$.

A core is said to be countable if $\mathrm{cor}(X)\le \omega$. The core of a discrete space is countable if and only if $X$ is countable.

Properties

• The core of any locally compact Lindelöf space is countable.
• If $X$ is locally compact with a countable core, then any closed discrete subset $H$ of $X$ is countable. That is the extent

$e(X)=\{Y:Y\text{ is a closed discrete subset of }X\}$

is countable.

• Locally compact spaces with countable core are σ-compact under a broad range of conditions.
• A subset $Y$ of $X$ is called compact from inside if every subset $F$ of $Y$ that is closed in $X$ is compact.
• A locally compact space $X$ has a countable core if there exists a countable open cover of sets that are compact from inside.