Why is cofinite topology not hausdorff?
SolitaryRoad.comWebsite owner: James Miller [ Home ] [ Up ] [ Info ] [ Mail ] Separation axioms. T1-Space. Cofinite topology. Hausdorff space. Regular and normal spaces. Urysohns Lemma and Metrization Theorem. Completely regular space. Tychonoff space. Separation axioms Def. T1-Space. A T1-space is a topological space X with the following property: 1] For any x, y ε X, if x y, then there is an open set that contains x and does not contain y. Syn. Frechet space Example 1. Consider the topological space X = {a, b, c} with a topology τ = {X, , {a, b}, {b}, {b, c} }. If we apply the test for a T1-space to elements a and b of this space we note that there is no open set in τ which contains a and does not contain b. Thus this space is not a T1-space. Example 2. Consider the real line R with the usual topology. For any two points x and y in R there is an open set that contains x and does not contain y. Thus the real line R with the usual topology is a T1-space. Theorem 1. A topological space X is a T1-space if and only if every singleton set {p} of X is closed. Def. Cofinite topology. Let X be a set and T be the collection of all subsets of X whose complements are finite, along with the empty set . Then T is a topology on X. It is called the cofinite topology on X. For example, let X be the real line R. The complement of, say, the finite set A = {3, 5, 8, 10} is R - A, an open set. It is the open set R with four discrete points missing. T consists of that class of sets in R characterized by the fact that they are complements of a discrete, finite set plus the empty set . In other words, it consists of: 1) all of those sets in R which have the property that their complements are finite sets, plus 2) the empty set . In general, each of the open sets of T consist of R minus a finite number of discrete points. The union and intersection of any two such sets will be another set of the same kind. Thus this class of sets form a topology for R. Because finite unions of closed sets are closed, Theorem 1 implies the following: Corollary 1. A topological space X with topology τ is a T1-space if and only if τ contains the cofinite topology on X. T1-topology. The cofinite topology on X is the coarsest topology on X for which X with topology τ is a T1-space . Consequently the cofinite topology is also called the T1-topology. Def. Hausdorff space (or T2-space). A Hausdorff space (or T2-space) is a topological space X with the following property: 2] If x y, then there are disjoint open sets U and V that contain x and y, respectively. See Fig. 1. Note that a Hausdorff space is always a T1-space. Theorem 2. Every metric space is a Hausdorff space. A T1-space that is not Hausdorff. The topological space consisting of the real line R with the cofinite topology, i.e. T1-topology, T is not Hausdorff. Proof. Let G and H be any non-empty open sets in T. We note that both G and H are infinite since they are complements of finite sets. If GH = , then G, an infinite set, would be contained in the finite complement of H. This is not possible so G and H cannot be disjoint. Thus no two distinct points in R belong to two disjoint open sets as required by the Hausdorff condition [T2]. Thus we see that T1-spaces need not be Hausdorff. In general, a sequence {a1, a2, a3, ...... }of points in a topological space X can converge to more than one point in X. This cannot happen if the space is Hausdorff. Theorem 3. If X is a Hausdorff space, then every convergent sequence in X has a unique limit. The converse of this theorem is not true without conditions. Theorem 4. Let X be first countable. Then X is Hausdorff if and only if every convergent sequence has a unique limit. Def. Regular space. A topological space X is said to be regular if for each pair consisting of a point x and a closed set B disjoint from x, there exist disjoint open sets containing x and B, respectively. See Fig. 2. Def. Normal space. A topological space X is said to be normal if for each pair A, B of disjoint closed sets of X, there exist disjoint open sets containing A and B, respectively. See Fig. 3. Theorem 5. A topological space X is normal if and only if for every closed set F and open set H containing F there exists an open set G such that Every metric space is normal. Def. T3-space. A regular T1-space. Def. T4-space. A normal T1-space. See Fig. 4. Urysohns Lemma and Metrization Theorem The following is the classical result of Urysohn. Urysohns lemma. If P and Q are two nonintersecting closed sets in a normal topological space T, then there exists a real function f defined and continuous in T and such that 0f(p)1 for all p, with f(p) = 0 for p in P, and f(p) = 1 for p in Q. James & James. Mathematics Dictionary Urysohns Metrization Theorem. A regular T1 topological space is metrizable if and only if it satisfies the second axiom of countability. Thus every metric space is a Lindelof space. Functions that separate points. Let A = { fi : i ε I } be a class of functions from a set X into a set Y. The class A of functions is said to separate points if and only if for any pair of distinct points a, b ε X there exists a function f in A such that f(a) f(b). Example. Consider the class of real-valued functions A = { f1(x) = sin x, f2(x) = sin 2x, f3(x) = sin 3x, ....... } defined on R. We observe that for every function fn ε A, fn(0) = fn(π) = 0. Thus none of the functions in A separate the points 0 and π (i.e. they all map the two points into the same point, rather than into separate points). Thus the class A does not separate points. Theorem 6. If the class C [X, R] of all real-valued continuous functions on a topological space X separates points, then X is a Hausdorff space. Proof Def. Completely regular space. A topological space X is said to be completely regular if for every point A of X and every closed set B not containing a, there exists a continuous function f : X [0, 1] such that f(a) = 0 and f[B] = 1. Theorem 7. A completely regular space is also regular. Def. Tychonoff space. A completely regular T1-space. Syn. T3 1/2-space A Tychonoff space is a T3-space and a T4-space is a Tychonoff space. Theorem 8. The class C [X, R] of all real-valued continuous functions on a completely regular T1-space X separates points References 1. Lipschutz. General Topology 2. Simmons. Introduction to Topology and Modern Analysis 3. Munkres. Topology, A First Course 4. James & James. 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