Compact Spaces with a $P$-base

TL;DR

This paper studies compact spaces with a $P$-base, proving under certain set-theoretic assumptions that such spaces are first-countable or countable, and constructs examples with specific properties using forcing.

Contribution

It establishes conditions under which compact spaces with a $P$-base are first-countable or countable, and constructs non-first-countable examples using forcing.

Findings

Compact spaces with an $oldsymbol{ ext{omega}^ ext{omega}}$-base are first-countable under $oldsymbol{ ext{omega}_1< ext{b}}$.

Scattered compact spaces with an $oldsymbol{ ext{omega}^ ext{omega}}$-base are countable under $oldsymbol{ ext{omega}_1< ext{b}}$.

Existence of non-first-countable compact spaces with a $P$-base in certain models.

Abstract

In the paper, we investigate (scattered) compact spaces with a $P$-base for some poset $P$. More specifically, we prove that, under the assumption $\omega_1<\mathfrak{b}$, any compact space with an $\omega^\omega$-base is first-countable and any scattered compact space with an $\omega^\omega$-base is countable. These give positive solutions to Problems 8.6.9 and 8.7.7 in \cite{Banakh2019}. Using forcing, we also prove that in a model of $\omega_1<\mathfrak{b}$, there is a non-first-countable compact space with a $P$-base for some poset $P$ with calibre~$\omega_1$.