Some Algebraic And Topological Properties Of The Completion Of A Certain Metric Space

Author

Billy Joe Maness, Prairie View Agricultural and Mechanical College

Date of Award

8-1969

Document Type

Thesis

Degree Name

Master of Science

Degree Discipline

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

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