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