Spaces with a $\mathbb{Q}$-diagonal

TL;DR

The paper proves that compact spaces with a $Q$-diagonal are metrizable and Tychonoff spaces with such diagonals are cosmic, resolving open problems in the field.

Contribution

It establishes that $Q$-diagonals imply metrizability in compact spaces and cosmicity in Tychonoff spaces, answering previously open questions.

Findings

Compact spaces with a $Q$-diagonal are metrizable

Tychonoff spaces with a $Q$-diagonal are cosmic

Provides positive solutions to open problems in topology

Abstract

A space $X$ has a $\mathbb{Q}$-diagonal if $X^2\setminus \Delta$ has a $\mathcal{K}(\mathbb{Q})$-directed compact cover. We show that any compact space with a $\mathbb{Q}$-diagonal is metrizable, hence any Tychonorff space with a $\mathbb{Q}$-diagonal is cosmic. These give a positive answer to Problem 4.2 and Problem 4.8 in \cite{COT11} raised by Cascales, Orihuela and Tkachuk.