Some properties of minimal S({\alpha}) and S({\alpha})FC spaces

TL;DR

This paper explores properties of minimal S(n) and S(n)FC spaces, characterizing various types of S(n)-spaces and answering open questions in the field.

Contribution

It provides new characterizations of S(n)-closed, S(n)-{ heta}-closed, and S(n)FC spaces using { heta}(n)-complete accumulation points, and introduces properties of R-closed and regular functionally compact spaces.

Findings

Characterized S(n)-closed and S(n)FC spaces via accumulation points.

Provided new properties of R-closed and regular functionally compact spaces.

Answered open questions in the theory of S(n)-spaces.

Abstract

A S(n)-space is S(n)-functionally compact (S(n)FC) if every continuous function onto a S(n)-space is closed. S(n)-closed, S(n)-{\theta}-closed, minimal S(n) and S(n)FC spaces are characterized in terms of {\theta}(n)-complete accumulation points. In paper we also give new characteristics of R-closed and regular functionally compact spaces. Results obtained to answer some questions raised by D.Dikranjan, E.Giuli, L.Friedler, M.Girou, D.Pettey and J.Porter.