Strongly Pseudoradial Spaces

TL;DR

This paper introduces strongly pseudoradial spaces, a new class of topological spaces that address limitations of weakly Hausdorff pseudoradial spaces by ensuring unique convergence properties and categorical closure.

Contribution

It develops the category of strongly pseudoradial spaces, proves it is Cartesian closed, and characterizes these spaces via unique extensions of maps from well-ordered spaces.

Findings

Strongly pseudoradial spaces form a coreflective hull of well-ordered spaces.

These spaces are Cartesian closed.

Analogues of sequential compactness and countable compactness are established.

Abstract

The "weakly Hausdorff" property for pseudoradial spaces fails to be naturally characterized by unique convergence of transfinite sequences. In response, we develop the category $\mathbf{SPsRad}$ of strongly pseudoradial spaces, compactly generated spaces whose closed sets are determined by globally continuous maps from well-ordered spaces. Categorically, $\mathbf{SPsRad}$ is the coreflective hull of the class of well-ordered spaces, and $\mathbf{SPsRad}$ is Cartesian closed. The strongly pseudoradial weakly Hausdorff spaces admit a natural characterization involving unique extensions of injective maps of well-ordered spaces. We also obtain analogs in $\mathbf{SPsRad}$ of the fact that for sequential spaces, sequential compactness is equivalent to countable compactness.