×

Cardinal properties of Hattori spaces and their hyperspaces. (English) Zbl 1331.54006

In [Mem. Fac. Sci. Eng., Shimane Univ., Ser. B, Math. Sci. 43, 13–26 (2010; Zbl 1196.54048)], Y. Hattori introduced a family, ordered by inclusion, of topologies on \(\mathbb{R}\), given by \(\mathcal{G} = \{\tau(A)\mid A \subseteq \mathbb{R}\}\) – where, for each \(A \subseteq \mathbb{R}\), the topology \(\tau(A)\) is defined as follows:
(1) For each \(x \in A\), \(\{]x - \varepsilon, x + \varepsilon[\) \(\mid \varepsilon > 0\}\) is a neighbourhood base at \(x\); and
(2) For each \(x \in \mathbb{R} \setminus A\), \(\{[x,x + \varepsilon[\) \(\mid \varepsilon > 0\}\) is a neighbourhood base at \(x\).
Notice that \(\tau(\emptyset) = \tau_S\) – the topology of the Sorgenfrey line, and \(\tau(\mathbb{R}) = \tau_E\) – the usual, Euclidean topology on \(\mathbb{R}\). Also, it is easy to see that \(A \supseteq B\) if, and only if, \(\tau(A) \subseteq \tau (B)\), and so \(\tau_E \subseteq \tau(A) \subseteq \tau_S\) for every \(A \subseteq \mathbb{R}\).
In the paper under review, the authors call the above described spaces Hattori spaces, and they investigate a number of cardinal properties of such spaces. More specifically, cardinal properties of a Hausdorff space \(X\) are compared with cardinal properties of its hyperspace – that is, the space of all non-empty closed subsets of \(X\) endowed with the Vietoris topology.

MSC:

54B20 Hyperspaces in general topology
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)

Citations:

Zbl 1196.54048
PDFBibTeX XMLCite