Functionally Hausdorff spaces. (English) Zbl 0689.18006

Lazowick Lord, Harriet

Cah. Topologie Géom. Différ. Catég. 30, No. 3, 247-256 (1989).

A space X is called functionally-Hausdorff provided that the set of continuous functions from X to the reals separates points in X. The author characterizes various types of special morphisms in the category FH of functionally Hausdorff spaces and characterizes the FH-epimorphisms and the FH-regular morphisms in TOP. It is shown that FH-epimorphisms are closed under composition but that FH-regular morphisms are not. A class of morphisms N is described for which (FH-epi,N) is a factorization structure on TOP. The main result is that there is no factorization structure (E,M) on TOP for which X is functionally Hausdorff iff the diagonal map \(\Delta_ X\) is in M.

Reviewer: G.E.Strecker

MSC:

18B30	Categories of topological spaces and continuous mappings (MSC2010)
18A20	Epimorphisms, monomorphisms, special classes of morphisms, null morphisms
54D10	Lower separation axioms (\(T_0\)–\(T_3\), etc.)

Keywords:

functionally Hausdorff spaces; epimorphisms; regular morphisms; factorization structure

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