Separation axioms. T1-Space. Cofinite topology. Hausdorff space. Regular and normal spaces. Urysohn�s Lemma and Metrization Theorem. Completely regular space. Tychonoff space.

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Separation axioms. T₁-Space. Cofinite topology. Hausdorff space. Regular and normal spaces. Urysohn’s 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 G H = ∅, 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.

Urysohn’s Lemma and Metrization Theorem

The following is the classical result of Urysohn.

Urysohn’s 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 0 f(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

Urysohn’s 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 = { f_i: 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 f_n ε A,

f_n(0) = f_n(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. T_{3 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. Mathematics Dictionary