Core of a locally compact space

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

The concept was introduced by Alexander Arhangel'skii.^[1][2]

Core of a locally compact space

Let ${\displaystyle X}$ be a locally compact and Hausdorff space. A subset ${\displaystyle S\subseteq X}$ is called saturated if it is closed in ${\displaystyle X}$ and satisfies ${\displaystyle S\cap P\neq \emptyset }$ for every closed, non-compact subset ${\displaystyle P\subseteq X}$.^[3]

The core ${\displaystyle \operatorname {cor} (X)}$ is the smallest cardinal ${\displaystyle \tau }$ such that there exists a family ${\displaystyle \gamma =(\gamma _{j})}$ of saturated subsets of ${\displaystyle X}$ satisfying: ${\displaystyle |\gamma |\leq \tau }$ and ${\displaystyle \bigcap _{j}\gamma _{j}=\emptyset }$.^[3]

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

Properties

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

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

is countable.

• Locally compact spaces with countable core are σ-compact under a broad range of conditions.^[4]
• A subset ${\displaystyle Y}$ of ${\displaystyle X}$ is called compact from inside if every subset ${\displaystyle F}$ of ${\displaystyle Y}$ that is closed in ${\displaystyle X}$ is compact.
• A locally compact space ${\displaystyle X}$ has a countable core if there exists a countable open cover of sets that are compact from inside.^[5]