Classification of topological spaces. II. (English) Zbl 0758.54001

Aull, Charles E.

Bull. Pol. Acad. Sci., Math. 36, No. 11-12, 649-654 (1988).

Summary: In part I [Bull. Acad. Pol. Sci., Ser. Sci. Math. Astr. Phys. 15, 773-778 (1967; Zbl 0171.434)], topological spaces were classified based on the number of distinct sets produced by means of closure and interior operators. This was inspired by the Kuratowski closure complement problem.
A space is an \(n\)-space if it has an \(n\)-set and no \(m\)-set for \(m>n\). A set \(M\) is an \(n\)-set if exactly \(n\) of the sets \(M\), \(M^ c\), \(M^{ci}\), \(M^{cic}\), \(M^ i\), \(M^{ici}\) are distinct. A space is a hereditary \(n\)-space if it has an \(n\)-subspace but no \(m\)-subspace for \(m>n\). Compact \(n\)-spaces are classified. A space with at least three nonisolated points is shown to be a hereditary 5-space iff it is either hereditarily extremally disconnected or hereditarily OU (OU = no open subset is resolvable), but not both.

MSC:

54A05	Topological spaces and generalizations (closure spaces, etc.)
54G05	Extremally disconnected spaces, \(F\)-spaces, etc.
54D30	Compactness

Keywords:

Kuratowski closure complement problem; hereditary \(n\)-space

Citations:

Zbl 0171.43403; Zbl 0171.434