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]

References

1. Arhangel'skii, Alexander (2007). "Locally compact spaces of countable core and Alexandroff compactification". Topology and Its Applications. 154 (3): 625–634. doi:10.1016/j.topol.2005.05.011. ISSN 0166-8641.
2. Tall, Franklin D. (2010). "On a core concept of Arhangel'skiĭ". Topology and Its Applications. 157 (8): 1541–1547. doi:10.1016/j.topol.2009.05.018.
3. Arhangel'skii 2007, p. 626.
4. Arhangel'skii 2007, pp. 627–628.
5. Tall 2010, p. 1541.