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 is countable. 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.
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