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.

Core of a locally compact space

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.

Properties

• 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.

References