Locally compact spaces of countable core and Alexandroff compactification (Q861944)

The author introduces a new cardinal invariant, the core of a locally compact \(T_2\)-space (denoted \(cor(X)\)), and states that ``locally compact spaces of countable core generalize locally compact \(\sigma\)-compact spaces in a way that is slightly exotic, but still quite natural''. A subset \(S\) of a locally compact space \(X\) is called saturated iff \(S\) is closed in \(X\) and \(S \cap P\not=\emptyset\), for every closed, non-compact subset \(P\) of \(X\). Then \(cor(X)\) is defined to be the smallest cardinal number \(\tau\) such that there exists a family \(\gamma\) of saturated subsets of \(X\) satisfying \( | \gamma | \leq\tau\) and \(\cap\gamma=\emptyset\). Among many results, the author shows that under a broad range of conditions locally compact spaces of countable core must be \(\sigma\)-compact. In particular, a locally compact space of countable core that is either normal or realcompact is \(\sigma\)-compact. The class of spaces of countable core is preserved in both directions by perfect mappings. The Alexandroff one-point compactification \(\alpha(X)=X\cup\{\infty\}\) of a non-compact locally compact space \(X\) is weakly first countable at \(\infty\) if and only if \(cor(X) = \omega\), and \(\alpha(X)\) is first countable (or Fréchet--Urysohn) at \(\infty\) if and only if \(X\) is \(\sigma\)-compact. The author shows that two spaces in the literature are non-\(\sigma\)-compact locally compact spaces of countable core. Eight problems are stated.