Core of a locally compact space

In topology, the core of a locally compact space is a cardinal invariant of a locally compact space, denoted by. Locally compact spaces with countable core generalize σ-compact locally compact spaces.
The concept was introduced by Alexander Arhangel'skii.

Let be a locally compact and Hausdorff space.
A subset is called saturated if it is closed in and satisfies
for every closed, non-compact subset.
The core is the smallest cardinal such that there exists a family of saturated subsets of satisfying:
and
A core is said to be countable if.
The core of a discrete space is countable if and only if is countable.

The core of any locally compact Lindelöf space is countable.
If is locally compact with a countable core, then any closed discrete subset of is countable. That is the extent
Locally compact spaces with countable core are σ-compact under a broad range of conditions.
A subset of is called compact from inside if every subset of that is closed in is compact.
A locally compact space has a countable core if there exists a countable open cover of sets that are compact from inside.