On cellularity, closure, sobriety and separation axioms in the lower vietoris topology.

Abstract

This thesis introduces four new classes of hyperspaces with the Lower Vietoris Topology, allowing for a complete study of various properties in these hyperspaces. One consists of those satisfying the Sub-base Condition, such as the hyperspace 2X of non-empty closed subsets of a topological space X. Another consists of those which are almost finitely natural, such as the hyperspace Fn(X) of n-element subsets of X, generalising the notion of natural families by Ivanova-Dimova. Interestingly, these hyperspaces are used to provide new characterisations of the sober T1 property in X as well as sobriety in the T0-identification of X. Sobriety and irreducibility are characterised in these hyperspaces in terms of X. It is shown that any pair of almost finitely natural hyperspaces have the same closure. A surprising relationship exists between irreducibility and closure for arbitrary hyperspaces. When X is Hausdor!, Fedorchuk showed that Fn(X) with the Vietoris Topology has the same cellularity as Xn and a subspace X[n]; Costantini et al. showed that the projection ˆj n : X[n] → Fn(X) is a local homeomorphism. This thesis shows that X[n] always has cellularity at most that of Xn and that X being Hausdor! is (almost always) equivalent to not only the quotient but also the perfect covering property in ˆjn. The quasi-open property of ˆjn is characterised in X. In this case, if X has infinite cellularity, then the cellularity of every almost finitely natural hyperspace is the same as Xn and X[n]. A non-Hausdor! space Y is constructed where ˆjn is quasi-open and the Vietoris Topology is not the same as the Lower Vietoris Topology on Fn(Y ). Known results on the T0 and T1 properties and the T0-identification in the Lower Vietoris Topology are extended to arbitrary hyperspaces. When X is T0, the TD property in 2X is characterised in terms of X as a corollary of a more general new result. The preregular, TD and T0 properties in Fn(X) are characterised in X; the latter two are shown to be equivalent in Fn(X). It is shown that Fn(X) is Hausdor! if and only if X is Hausdor!