Connectedness properties of the space of closed subsets of a topological space.

Abstract

Given a topological space X we denote by F(X) the class of all closed subsets of X and by Fo(X) the class of all nonempty closed subsets of X. We place a topology on these sets called the topology of closed convergence tauc. In this thesis, we are mainly concerned with the connectedness properties of the spaces F(X) and Fo(X). Our work is divided into four chapters. In chapter I, we introduced a notion of convergence for a net of subsets and we proved some properties of the topological limits which are used throughout the thesis. We also defined the topology of closed convergence and proved that (F(X), tauc ) is compact and if X is locally compact then the topology of closed convergence induces the topological convergence of nets of sets, but if X is not locally compact, then there exists no topology on F(X) which induces the topological convergence of nets of sets. In chapter II, we proved that F and Fo are functors from the category of compact Hausdorff spaces to itself which preserve the homotopy relation, we show that F is not a functor if X is not compact; however Fwc (X) is a functor. In chapter III, we studied the connectedness properties of the spaces F(X) and Fo(X), and we proved that Fo(C) ≃ C if C is the Cantor space and Fo(X) is contractible if X is a finite connected simplicial complex. We also proved that if X is connected then Fo(X) is connected and if X is not compact, F(X) is connected. In Chapter IV, we studied the inverse limit space Finfinityo (X) and we proved that Fo( lim← Xlambda) ≃ lim← Fo (Xlambda) for a net of compact Hausdorff spaces and Fo(X) is contractible if X is contractible, we also proved the continuity of the two maps &phis; (X) : X→Fo (X) and F(X) : F2o (X) → Fo (X), and finally we proved that Fo( Finfinityo (X))≃ Finfinityo (X).