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Zbl 0891.18004
Dikranjan, Dikran; Pelant, Jan
Categories of topological spaces with sufficiently many sequentially closed spaces. (English)
[J] Cah. Topologie Géom. Différ. Catég. 38, No. 4, 277-300 (1997). ISSN 0008-0004

For suitable full subcategories ${\cal A}$ of ${\cal T}op$, the authors investigate the concepts (a) of sequentially closed ${\cal A}$-objects and (b) of ${\cal A}$ having enough sequentially closed objects. Typical results: 1. The sequentially closed objects in $T_1$ (resp. ${\cal T}ych$, ${\cal U} {\cal S}=$ spaces with unique sequential limits, ${\cal S} {\cal U} {\cal S}=$ spaces in which convergent sequences have a unique accumulation point) are the finite (resp. countably compact, sequentially compact, countably compact) objects. 2. ${\cal T}ych$ and ${\cal R}eg$ do not have enough sequentially closed objects [for ${\cal R}${\it eg} this has originally been shown by {\it I. Gotchev} in Rend. Ist. Mat. Univ. Trieste 20, No. 1, 1-17 (1988; Zbl 0689.54013)]. 3. ${\cal U} {\cal S}$ and ${\cal S} {\cal U} {\cal S}$ have enough sequentially closed objects.
[H.Herrlich (Bremen)]

MSC 2000:: *18B30 Categories of topological spaces
54D25 "P-minimal" and "P-closed" spaces
54B30 Categorical methods in general topology
54D35 Compactifications

Keywords: sequentially closed objects

Citations: Zbl 0689.54013

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