#### Date of Award

8-1969

#### Document Type

Thesis

#### Degree Name

Master of Science

#### Department

Master of Mathematics

#### Abstract

The primary objective of this paper is to consider a metric space (X,d) that is not complete and analyze the problem of enlarging the set X and extending the metric d to obtain a new metric space ^{(X*d*)} that is complete and that contains (X,d) as a dense subspace. To solve this problem we must provide some new points to serve as limit points for the nonconvergent Cauchy sequences of (X,d), and we must extend the metric d to these new points in such a way that we are assured that the formerly nonconvergent Cauchy sequences will find limits among the new points.

Chapter I gives us all the symbols, definitions, and auxiliary theorems that are needed in order to work with metric spaces within the scope of this paper. Since this paper is dealing with a general topological space, most of our theorems and definitions will be stated in terms of a metric space. Chapter II is composed of three main theorems which make up the body of the paper. Theorem I deals with building up a metric space as described in the paragraph above. Since this theorem is very involved, we will use a series of Lemmas to arrive at the proof of Theorem I. Theorem 2 take the function f as described in Lemma 1.4 and show that f ( X ) is dense in X . Theorem 3 is another property that relates the two metric spaces (X,d) and (X ,d ). It states that the metric space.

(X*d*) is separable if and only if (X,d) is separable. Chapter III is devoted to an application of the paper itself. It will show how we can construct the set of real numbers from the set of rationals.

#### Committee Chair/Advisor

E. E. Thornton

#### Committee Member

A. D. Stewart

#### Publisher

Prairie View Agricultural and Mechanical College

#### Rights

© 2021 Prairie View A & M UniversityThis work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

#### Date of Digitization

3/31/2022

#### Contributing Institution

John B Coleman Library

#### City of Publication

Prairie View

#### MIME Type

Application/PDF

#### Recommended Citation

Maness, B. J. (1969). Some Algebraic And Topological Properties Of The Completion Of A Certain Metric Space. Retrieved from https://digitalcommons.pvamu.edu/pvamu-theses/1430