All spaces of countable spread can be small

TL;DR

This paper proves that under certain set-theoretic assumptions, all spaces with countable spread are small in size, specifically at most continuum, and characterizes their open set structure.

Contribution

It establishes the consistency, relative to a weakly compact cardinal, of a set-theoretic and topological property linking partition relations to bounds on space size.

Findings

Spaces of countable spread have size at most continuum under the assumptions.

The paper links partition properties with topological space size constraints.

It characterizes the open set structure of such spaces.

Abstract

The main result of this paper is the proof of the simultaneous consistency, modulo a weakly compact cardinal, of the equality $2^{< \mathfrak{c}} = \mathfrak{c}$ with the following property (*) of partitions of pairs of $\mathfrak{c}$: \smallskip (*) For any coloring (or partition) $k : [\mathfrak{c}]^2 \rightarrow 2$ either there is a homogeneous set of size $\mathfrak{c}$ in color $0$ or there is a set $S \in [\mathfrak{c}]^\mathfrak{c}$ such that for every countable $A \subset S$ there is $\beta \in \mathfrak{c}$ for which $A \subset \beta$ and $k(\{\alpha, \beta\}) = 1$ for all $\alpha \in A$. \smallskip (*) plus $2^{< \mathfrak{c}} = \mathfrak{c}$ together then imply that for every topological space $X$ of countable spread, i.e. not containing any uncountable discrete subset, $|X| \le \mathfrak{c}$ if it is Hausdorff and $o(X) = \mathfrak c$ if it is also infinite and regular. Here $o(X)$ denotes the number of all open subsets of $X$.