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Finite Topological Spaces University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 8-2013 Finite Topological Spaces Jimmy Edward Miller [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Part of the Geometry and Topology Commons Recommended Citation Miller, Jimmy Edward, "Finite Topological Spaces. " Master's Thesis, University of Tennessee, 2013. https://trace.tennessee.edu/utk_gradthes/2438 This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by Jimmy Edward Miller entitled "Finite Topological Spaces." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Master of Science, with a major in Mathematics. James Conant, Major Professor We have read this thesis and recommend its acceptance: Conrad Plaut, Charles Collins Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) University of Tennessee, Knoxville Trace: Tennessee Research and Creative Exchange Masters Theses Graduate School 8-2013 Finite Topological Spaces Jimmy Edward Miller [email protected] This Thesis is brought to you for free and open access by the Graduate School at Trace: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of Trace: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by Jimmy Edward Miller entitled "Finite Topological Spaces." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Mathematics. James Conant, Major Professor We have read this thesis and recommend its acceptance: Conrad Plaut, Charles Collins Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official student records.) Finite Topological Spaces A Thesis Presented for the Master of Science Degree The University of Tennessee, Knoxville Jimmy Edward Miller August 2013 c by Jimmy Edward Miller, 2013 All Rights Reserved. ii I dedicate this paper to my fiance Jillian. iii Acknowledgements I would like to thank God, Jillian, Dr. Conant, Jessie, and my parents for their support and guidance. iv Go down deep enough into anything and you will find mathematics. Dean Schlicter v Abstract A thesis on some fundamental aspects of Algebraic Topology on Finite Topological Spaces, specifically 4 point spaces. vi Table of Contents 1 Introduction1 2 Continuous Functions and Separation Axioms4 3 Connected and Path Connected Spaces8 4 Four Point Spaces 13 5 Algebraic Topology on Finite Spaces 16 Bibliography 21 Vita 23 vii Chapter 1 Introduction We begin by giving a brief overview of the fundamentals of point-set Topology. A basic understanding of set arithmetic will be assumed. These fundamentals will provide an adequate foundation to understand the topics in Chapters 4 and 5. Definition 1.0.1. A topology on a set X is a collection T of subsets of X having the following properties: (i) ;;X 2 T [ (ii) Given a collection fAαgα2Ω where Aα 2 T , then Aα 2 T . α2Ω n n \ (iii) Given a finite collection fAigi=1 where Ai 2 T , then Ai 2 T . i=1 Munkres(2000) Elements of T are called open sets. The complement of an open set is called a closed set. Sets that are both open and closed are called clopen sets. Definition 1.0.2. Let (X; T ) be a topological space and A ⊆ X. The subspace topology of A is the collection TA = fA \ U where U 2 T g. Munkres(2000) Let's verify that TA is in fact a topology. (i) A = A \ X 2 TA and ; = A \ ; 2 TA since X; ; 2 T 1 [ (ii) Let fWαgα2Ω ⊆ TA. Then Wα = A \ Uα where Uα 2 T . Then Wα = α2Ω [ [ (A \ Uα) = A \ ( Uα) 2 TA α2Ω α2Ω n n \ (iii) Let fWigi=1 2 TA. Then Wi = A \ Ui where Ui 2 T . Then Wi = i=1 n n \ \ (A \ Ui) = A \ ( Ui) 2 TA. i=1 i=1 Hence by Definition 1.0.1, TA is a topology. Definition 1.0.3. The discrete topology on a set X is the topology that includes all subsets of X, i.e the power set of X. The coarse topology on X is the topology consisting of only X and ;. The coarse topology is also called the trivial topology. Definition 1.0.4. Let X be a set. A basis for a topology on X is a collection of subsets B of X called basis elements such that the following properties hold: (i) For every x 2 X there exists some B 2 B containing x. (ii) If x 2 B1 \ B2 for some B1;B2 2 B, then there exists some B3 2 B where x 2 B3 ⊆ B1 \ B2. Munkres(2000) If such a collection B satisfies these two properties, we can define the topology T generated by B as follows: A set U is open in X if for every x 2 U there exists some B 2 B where x 2 B ⊆ U. Let's verify that T is a topology on X. First, for any x 2 X we know by (i) that there exists some B 2 B containing x; and since B ⊆ X be definition, we know that X is open. Also, ; is open vacuously. [ Let fUαgα2Ω be open in X. Let U = Uα. Suppose that x 2 U. Then x 2 Uβ for α2Ω some β 2 Ω. Since Uβ is open we know there exists B 2 B such that x 2 B ⊆ Uβ ⊆ U. Hence U is open. n n \ Let fUigi=1 be open in X. Let V = Ui. Suppose that x 2 V . Then x 2 Ui i=1 for all i. Hence there exists B 2 B where x 2 B 2 Ui for all i. Which implies that B ⊆ V , hence V is open. Therefore T is a topology. 2 Corollary 1.0.1. Every open set is a union of basis elements Proof. Let (X; T ) be a topological space where B is a basis for T . Let U ⊆ X be open. [ Then for every x 2 U there exists Bx 2 B where x 2 Bx ⊆ U. Hence Bx = U. x2U Lemma 1.0.1. A set U ⊆ X is open if and only if for every x 2 U there exists an open Vx ⊆ X where x 2 Vx ⊆ U. Proof. Let U ⊆ X and x 2 U. Suppose there exists an open Vx ⊆ X where x 2 Vx ⊆ [ U. It is clear that Vx ⊆ U since Vx ⊆ U for all x 2 X. Also, for any x 2 U, x2U [ [ x 2 Vx, hence x 2 Vx. Therefore U = Vx. Hence U is a union of open sets, x2U x2U hence open. Definition 1.0.5. Let X and Y be sets. Define X t Y to be the disjoint union of X and Y , i.e X \ Y = ;. Definition 1.0.6. Let (X; T ) and (Y; Γ) be topological spaces. Then the topology of X t Y consists of sets of the form U t V where U is open in X and V is open in Y . 3 Chapter 2 Continuous Functions and Separation Axioms Continuous functions play a major part in Topology. We will use continuous functions and their properties extensively in Chapter 5 to prove properties of some finite topological spaces. Separation axioms give a method of classifying topological spaces given certain characteristics. Definition 2.0.7. Let (X; T ) and (Y; Γ) be topological spaces. A function f : X ! Y is called continuous if for every open U ⊆ Y , f −1(U) is open in X. Munkres(2000) Lemma 2.0.2. Let (X; T ) and (Y; Γ) be topological spaces. f : X ! Y is continuous if and only if given open U ⊆ Y , for every f(x) 2 U there exists open V ⊆ X containing x where f(V ) ⊆ U. Proof. ()) Let U ⊆ Y be open and f(x) 2 U. Since f is continuous f −1(U) is open. Hence there exists an open neighborhood V such that x 2 V ⊆ f −1(U), i.e f(V ) ⊆ U. (() Let U ⊆ Y be open. We know that for every f(x) 2 U there exists open V containing x such that f(V ) ⊆ U, i.e x 2 V ⊆ f −1(U). Hence f −1(U) is open. Therefore f is continuous. 4 Lemma 2.0.3. Let (X; T ) and (Y; Γ) be topological spaces. A function f : X ! Y is continuous if and only if for closed C ⊆ Y , f −1(C) is closed in X Proof. ()) Let C ⊆ Y be closed. Then Y nC is open. Hence f −1(Y nC) = f −1(Y )nf −1(C) = Xnf −1(C) is open. Therefore f −1(C) is closed. (() Let U ⊆ Y be open. Then Y nU is closed. Hence f −1(Y nU) = f −1(Y )nf −1(U) = Xnf −1(U) is closed.
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