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The purpose of this thesis is to study some of the properties of metric spaces. An effort is made to show that many of the properties of a metric space are generalized properties of R, the set of real numbers, or Euclidean n--space, and are specific cases of the properties of a general topological space.

In the study of groups and topological spaces, the properties of both are often encountered in one system. The following are common examples: groups with discrete topologies, the complex numbers with the usual topology, and matrix groups with metric topologies. The need for a study of how algebraic properties and topological properties affect one another when united and interrelated in one system soon becomes evident. Thus the purpose of this thesis is to study the interrelated group and topological space, the topological group.

This paper will be devoted to an exposition of some of the basic properties of paracompact spaces. In particular, it will be shown that every pseudo-metrizable space is paracompact and countably paracompact.

The purpose of this thesis is to investigate the idea of topological "connectedness" by presenting some of the basic ideas concerning connectedness along with several related concepts.

The study of metric spaces is closely related to the study of topology in that the study of metric spaces concerns itself, also, with sets of points and with a limit point concept based on a function which gives a "distance" between two points. In some topological spaces it is possible to define a distance function between points in such a way that a limit point of a set in the topological sense is also a limit point of the same set in a metric sense. In such a case the topological space is "metrizable". The real numbers with its usual topology is an example of a topological space which is metrizable, the distance function being the absolute value of the difference of two real numbers. Chapters II and III of this thesis attempt to classify, to a certain extent, what type of topological space is metrizable. Chapters IV and V deal with several properties of metric spaces and certain functions of metric spaces, respectively.

Cheng Hon-wing. / Thesis (M.Phil.)--Chinese University of Hong Kong. / Bibliography: leaf 29.

Hung Cheung Yan. / Thesis (M.Phil.)--Chinese University of Hong Kong. / Bibliography: leaves 28-29.