Hyperspace Topology And Continuous Selections

The study of hyperspace is from the idea of topologizing the collection of closed subsets of a space X. When these collections are topologized, they are called hyperspaces of X. And the mostly discussed collection is the family of all non-empty closed subsets of space X, denoted by F(X). The first step toward topologizing F(X) was taken by F.Hausdorff [8], who defined a metricρ_H on F(X), later called the Hausdorff metric, in the case when X is a bounded metricspace as follows: For A,B∈F(X)ρ_H (A, B) = max whereρis a metric on X. However because of the limit of the definition, the research did not become popular. In 1923, L.Vietoris [16] defined a hyperspace topology for a normal space X which is called Vietoris topology (or finite topology). In 1951, E.Michael identified the basic neighborhoods of Vietoris topology in his famous paper "Topologies on spaces of subsets" [6]. And he also introduced the definition of continuous selection on hyperspaces. In 1956, E.Michael in the papers [2, 3, 4] continued his study of continuous selections especially in the extension of continuous selections which he called as "selection problem". These papers marked the beginning of selection theory. E.Michael's work inspired the more enthusiasm of topologists. They defined various topologies on the hyperspaces and discussed the relation of continuous selection corresponding to these topologies and the properties of space X. With the study of selection theory becomeing more and more deeply and systemized, the selection theory gradually becomes an independent branch of general topology.The main themes in studying hyperspaces of a space X are summarized in the following two points:(1) to investigate what properties of X are carried over to F(X), C(X) or 2^X.(2) to determine whether F{X), C{X) and 2^X belong to C or not if X belongs to C for a class C of generalized metric spaces or spaces with some special properties.Some basic properties such as compactness, connectedness, separation axioms were studied by Michael [6]. And he also gave the definition of continuous selections. After Michael's work [2, 3, 4], the relationship between continuous selections and the properties of space X and the extension problem of selections become the main research directions.Recently,S. Garcia-Ferreira, V. Gutev and T. Nogura [2006] [24] continued the work of Michael. They gave a condition for the extension of continuous selection from F₂(X) to F₃(X).In Chapter 1, we mainly discussed the relation between continuous selections and the properties of topological space X. There are three aspects of the main results. Firstly in section 1.3, we studied the relationship between connectedness of space X and continuous weak selections, and proved if X has exactly one continuous weak selection, then X must be connected. This result weakened the hypothesis in the relative theorem proved by T.Nogura and D.Shakhmatov. In section 1.4, we pointed out that there was a gap in the proof of Theorem 3.1 [30] and gave a corrected proof. In section 1.5, we prove that if space X hasÏ„_V continuous zero-selections, then the cardinality of X is equal to the cellularity. This is an improvement of the result proved by Giuliano Artico, Umberto Marconi and Jan Pelant [12].In Chapter 2, the extension of continuous weak selection is discussed. In section 1.2, we introduced the definition of f-maximum and f-minimum. In section 1.3, the existence and some basic properties of f-maximum and f-minimum are discussed. In section 1.4, we prove that if p∈X and there exists a continuous selection on K.(X \ {p}), then there exists a continuous selection on K(X). This result is an improvement of the result of S. Garcia-Ferreira, V. Gutev and T. Nogura [12]. In section 1.5, we give an application of our main theorem on hereditarily paracompact scattered space. This theorem is a partial answer of [28, Question 5]. Finally we give a counterexample about the main theorem of this section.