×

Hyperspace of finite unions of convergent sequences. (English) Zbl 1499.54037

If \(X\) is a topological space the hyperspace consisting of all convergent sequences in \(X\) and as well as all their finite unions is denoted by \(\mathcal{S}(X)\) and endowed with the Vietoris topology.
In the first part of the paper the authors give an internal characterisation of the convergent sequences in \(\mathcal{S}(X)\). Then they compare various cardinality properties of \(X\) with those of \(\mathcal{S}(X)\). These are e.g. the character, the pseudocharacter, and on the other hand the minimal cardinality of certain networks on \(X\).
Furthermore they give an example of a topological space \(X\) which has a rank-2-diagonal such that \(\mathcal{S}(X)\) does not have a G\(_\delta\)-diagonal and show that \(\mathcal{S}(X)\) is a \(\gamma\)-space if and only \(X\) has this property.

MSC:

54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
40B99 Multiple sequences and series
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54B20 Hyperspaces in general topology
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D25 “\(P\)-minimal” and “\(P\)-closed” spaces
54E20 Stratifiable spaces, cosmic spaces, etc.
PDFBibTeX XMLCite
Full Text: DOI arXiv