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Zbl 1050.54001
Bella, Angelo; Simon, Petr
Spaces which are generated by discrete sets. (English)
[J] Topology Appl. 135, No. 1-3, 87-99 (2004). ISSN 0166-8641

The systematic research of the property of a space to be generated by its discrete subsets was started out by {\it A. Dow, M. G. Tkachenko}, {\it V. V. Tkachuk} and {\it R. G. Wilson} [Glas. Mat., III. Ser. 37(57), No. 1, 187--210 (2002; Zbl 1009.54005)] who among other things proved that sequential spaces, monotonically normal spaces and compact countably tight spaces are discretely generated. In the paper under review the authors continue this study. Among other interesting results they show the following: Theorem 1. Every Hausdorff space of countable fan tightness is discretely generated; Theorem 2. Every countable compact regular space of countable tightness is discretely generated. Theorem 1 and the well-known result saying that $C\sb{p}(X)$ has countable fan tightness if and only if $X\sp{n}$ is a Hurewicz space for each $n<\omega$ permit the authors to obtain two appealing consequences: (1) if $X$ is a $\sigma$-compact space, then $C\sb{p}(X)$ is discretely generated, and (2) every normed linear space in the weak topology is discretely generated. The last section of the paper contains two interesting non-trivial examples: (1) Under [CH], there exists a pseudoradial compact zero-dimensional space which is not discretely generated, and (2) Under [CH], there is a pseudocompact Tychonoff space of countable tightness which is not discretely generated. Notice that both examples need the Continuum Hypothesis. It is not known if they are true in ZFC.
[Manuel Sanchis (Castelló)]

MSC 2000:: *54A05 Topological spaces and generalizations
54H11 Topological groups (topological aspects)
54C10 Special maps on topological spaces
22A05 Structure of general topological groups

Keywords: discretely and weakly discretely generated sets; fan tightness; countable compactness; pseudoradial spaces; remote point; pseudocompactness; Lindelöff P-spaces

Citations: Zbl 1009.54005

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