D-4 - Pseudoradial Spaces

Abstract

The class of pseudo radial spaces, with the name of folgenbestimmte Räume, was introduced by H. Herrlich in 1967; they have also been called chain-net spaces. They are defined using the convergence of transfinite sequences. A (transfinite) sequence is a map whose domain is an ordinal. In these cases, only regular initial ordinals need to be considered, that is, regular cardinal numbers. Some subclasses of the class of pseudo radial spaces were introduced successively because they were useful for solving natural problems that arise for pseudo radial spaces. The class of almost radial spaces came first. Herrlich proved that the pseudo radial spaces are quotients of linearly ordered topological spaces and are quotients of spaces in which every point has a well ordered local base of neighborhoods. All Fréchet–Urysohn spaces are sequential and radial. All radial and all sequential spaces are almost radial, and all almost radial spaces are pseudo radial. One of the first problems considered in connection with the notion of pseudo radialness was to find Zermelo–Fraenkel set theory with the axiom of choice (ZFC) examples of non-sequential pseudo radial spaces with countable tightness. It is evident from Balogh's theorem that compact spaces of countable tightness are sequential.