All spaces of countable spread can be small

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) $k:[\mathfrak{c}{]}^2\to 2$ 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 ${\hat{I}}^2\in \mathfrak{c}$ for which $A\subset {\hat{I}}^2$ and $k(\{\hat{I}\pm ,{\hat{I}}^2\})=1$ for all $\hat{I}\pm \in A$. \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 $X$.

13 pages