US-spaces and closure operators. (English) Zbl 0599.54016

Tozzi, Anna

Suppl. Rend. Circ. Mat. Palermo, II. Ser. 12, 291-300 (1986).

A categorical space is a US space if every convergent sequence has a unique limit. It is SUS space if every convergent sequence has a unique accumulation point. It is known that \(T_ 2\Rightarrow SUS\Rightarrow US\Rightarrow T_ 1\), but that if the spaces are not first countable each implication is non-reversible. S. Salbany [Categorical Topology, Proc. Conf. Mannheim 1975, Lect. Notes Math. 540, 548-565 (1976; Zbl 0335.54003)] introduced a closure operator \([.]_ A\) associated with a reflective subcategory A of Top. The A-closure of a subset M of X is given by \([M]_ A=\cap \{K(f,g)| K(f,g)\supset M\), f,g: \(X\to Y\), \(Y\in A\}\) where \(K(f,g)=\{x\in X|\) \(f(x)=g(x)\}\). In this paper a characterization of US closed and SUS closed sets is given and it is proved that the category SUS is cowell-powered.

Reviewer: T.Porter

MSC:

54B30	Categorical methods in general topology
54A05	Topological spaces and generalizations (closure spaces, etc.)
54D10	Lower separation axioms (\(T_0\)–\(T_3\), etc.)

Citations:

Zbl 0335.54003