Measures on Suslinean spaces

TL;DR

This paper investigates the existence of certain non-separable compact spaces supporting measures, demonstrating their existence under Martin's axiom and non-existence in the random model, highlighting the influence of set-theoretic assumptions.

Contribution

It establishes the conditions under which non-separable compact measure-supporting spaces with countable π-character exist or do not exist, depending on set-theoretic axioms.

Findings

Existence of such spaces under Martin's axiom.

Non-existence of such spaces in the random model.

Spaces cannot be mapped onto $[0,1]^{oldsymbol{ ext{omega}}_1}$.

Abstract

We study the existence of non-separable compact spaces that support a measure and are small from the topological point of view. In particular, we show that under Martin's axiom there is a non-separable compact space supporting a measure which has countable $\pi$-character and which cannot be mapped continuously onto $[0,1]^{\omega_1}$. On the other hand, we prove that in the random model there is no non-separable compact space having countable $\pi$-character and supporting a measure.