Student Work

8-1-1972

Thesis

Degree Name

Master of Arts (MA)

Department

Mathematics

Dr. J. Scott Downing

Abstract

For an arbitrary topological space algebraic topology prescribes a construction for a fundamental group. There is a natural way of imposing a topology on this set. We will examine this construction and the topological space which results. We use the following notation and definitions: 1. I f X and Y are topological spaces and ACX, BCY and if f:X+Y is a map, we write f: (X,A)➔(Y,B) if f(A)cB. 2. (Y,B)CX,A)={f: (X,A)+(Y,B)I f is continuous}. 3. 1=[0,1], the closed unit interval; i={O,l}, the points O and 1. 4. If X is a topological space and a£X, then R is an equivalence relation on (X,a) ( I ,I ) defined as follows: If f,gE(X,a)(I, i ) then fRg if and only if f and_ g are homotopic relative to I, written f��g rel I. That is, there exists a continuous function F: I xl+X such that F(O,t)= f(t), F(l,t)=g(t), F(x,O)=a, F(x,l)=a for all x,tsl. R is easily shown to be an equivalence relation, [2, p. 6]. We write Rf��{g£(X,a)(I,I )I gRf}. 5. When we speak of a topology on (Y,B) (X,A) we use the "compact-open" topology. The compact-open topology has subbasic sets of the form (C,U) where CCX i�� compact, UC"f is open, and (C,U)��{fE(Y,_B)(X, A) I f(C)cU} . 6. Q(X,a)=(X,a) (I ,I ) with the compact-open topology is called the loop space of X at a.

A Thesis Presented to the Department of Mathematics and the Faculty of the Graduate College University of Nebraska In Partial Fulfillment of the Requirements for the Degree Master of Arts University of Nebraska at Omaha. Copyright 1972 Daniel William Atkinson

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