Weakly Corson compact trees

TL;DR

This paper introduces a new topology on trees called the countably coarse wedge topology, which makes certain trees into countably compact, Fréchet–Urysohn spaces, and explores its implications for weakly Corson and Valdivia compacta.

Contribution

It defines a novel topology on trees, demonstrates its properties, and provides the first example of a compact space with all closed subspaces weakly Valdivia but not weakly Corson.

Findings

The countably coarse wedge topology makes trees into countably compact, Fréchet–Urysohn spaces.

It establishes the role of this topology in weakly Corson and Valdivia compacta.

Provides the first example of a compact space with all closed subspaces weakly Valdivia but not weakly Corson.

Abstract

We introduce and study a new topology on trees, that we call the countably coarse wedge topology. Such a topology is strictly finer than the coarse wedge topology and it turns every chain complete, rooted tree into a Fr\'echet--Urysohn, countably compact topological space. We show the r\^{o}le of such topology in the theory of weakly Corson and weakly Valdivia compacta. In particular, we give the first example of a compact space $T$ whose every closed subspace is weakly Valdivia, yet $T$ is not weakly Corson. This answers a question due to Ond\v{r}ej Kalenda.