Tightness, character and related properties of hyperspace topologies

Abstract

Given a Hausdorff space X, we calculate the tightness and the character of the hyperspace CL∅(X) of X, endowed with either the co-compact or the lower Vietoris topology, and give some estimates for the tightness of CL∅(X), endowed with the Fell topology.\ud \ud Some properties related to first-countability and countable tightness, such as sequentiality, Fréchet property and, less directly, radiality and pseudoradiality, are investigated as well.\ud \ud To carry out our investigation, we also consider on the base space X several cardinal functions, and we compare some of them (which are newly defined or not so well known) with other classical ones, obtaining results and counterexamples which may be of some independent interest

Complete systems, elementary submodels and the tightness of upper hyperspaces

“…2 Indeed, on the one hand, this will cause no real loss of generality, because of the equality (that we have just pointed out): τ + Δ = τ + Δ∪{∅} for any Δ ⊆ CL ∅ (X). On the other hand, assuming ∅ ∈ Δ allows us to write a generic τ + Δ -basic neighbourhood of a given C ∈ CL ∅ (X) in the form (D c 1 ) + ∩ · · · ∩ (D c n ) + , where D 1 , .…”

“…In a recent paper [2], Costantini, Holá and Vitolo work with the least infinite cardinality of a complete system; consistently with the result of Frolík, they call such a cardinal function theČech number (see also [1]). …”

“…In [2], the authors give some estimates for the tightness of CL ∅ (X), the set of all closed subsets of a topological space X, endowed with the co-compact topology. They show in particular that such a cardinal function may be expressed as the open hereditary version of the k-Lindelöf number of the base space X; as a consequence, they obtain some bounds on the tightness of co-compact hyperspaces of regular spaces, namely the product of theČech number of X with the hereditary Lindelöf number of X.…”

Fell topology on the hyperspace of a non-Hausdorff space

“…Most of the results of this section are already known in the case in which X is a Hausdorff space-see [3,5,6,12,14].…”

The Reznichenko property and the Pytkeev property in hyperspaces

“…According to a result from [18], each compact Hausdorff space of countable tightness is a Pytkeev space. On the other hand, in [12] (see also [4]) it was shown that for a locally compact Hausdorff space X the tightness of (2 X , F) is countable if and only if X is hereditarily separable and hereditarily Lindelöf. So we have:…”