Subspaces of pseudoradial spaces

TL;DR

This paper demonstrates that all T0 and T1 topological spaces can be embedded into pseudoradial spaces, resolving a longstanding problem and characterizing the minimal coreflective subcategory of topological spaces.

Contribution

It proves the embeddability of T0 and T1 spaces into pseudoradial spaces and identifies the smallest coreflective subcategory with this property.

Findings

Every T0 and T1 space can be embedded in a pseudoradial space.

Characterization of the smallest coreflective subcategory of Top.

Resolution of a problem posed in earlier research.

Abstract

We prove that every topological space (T0-space, T1-space) can be embedded in a pseudoradial space (in a pseudoradial T0-space, T1space). This answers the Problem 3 in [Arhangelskii, A.V. - Isler, R. - Tironi, G: On pseudo-radial spaces, Comment. Math. Univ. Carolin. 27 (1986), 137-156]. We describe the smallest coreflective subcategory A of Top such that the hereditary coreflective hull of A is the whole category Top. (The same result without any separation axiom was proved already in the master thesis E. Murtinov\'a: Podprostory pseudoradi\'aln\'ich prostor\r{u}. I wasn't aware of this work when preparing this paper.)