Boundedness in topological spaces. (English) Zbl 1199.54030

Di Maio, Giuseppe; Kočinac, Ljubiša D. R.

Mat. Vesn. 60, No. 2, 137-148 (2008).

A family \(\mathbb B\) of nonempty closed subsets of a space is said to be an (abstract) boundedness if it is closed for finite unions, closed hereditary and contains all singletons. (Compare [S. T. Hu, J. Math. Pures Appl., IX. Sér. 28, 287–320 (1949; Zbl 0041.31602)].)
In the third section of the paper, several theorems state the equivalence of covering properties of spaces defined in terms of selection principles \({\mathbf S}_1({\mathcal A},{\mathcal B})\), \({\mathbf S}_{fin}({\mathcal A},{\mathcal B})\), \({\mathbf U}_{fin}({\mathcal A},{\mathcal B})\), \(\alpha_i({\mathcal A},{\mathcal B})\), \(i=2,3,4,\) for \(\mathbb B\)-covers, \(\gamma\)-covers and \(\gamma_{\mathbb B}\)-covers, and their game-theoretical and Ramsey-theoretical equivalents.
In the last two sections, relations between some covering properties of a space \(X\) and countable (strong) fan tightness of the function space \(C_b(X)\), respectively, the corresponding covering properties of the hyperspace \(\mathbb B\) equipped with the upper Vietoris topology, are established.
The statements generalize the already known results for special classes of subsets (closed, finite, compact).

Reviewer: Mila Mršević (Beograd)

MSC:

54A25	Cardinality properties (cardinal functions and inequalities, discrete subsets)
54C35	Function spaces in general topology
03E02	Partition relations
91A44	Games involving topology, set theory, or logic
54B20	Hyperspaces in general topology

Keywords:

\(\mathbb B\)-cover; \(\gamma_{\mathbb B}\)-cover; selection principles; topological games; countable strong fan tightness; \(C_b(X)\); hyperspaces

Citations:

Zbl 0041.31602