SolitaryRoad.com

Website owner: James Miller

[ Home ] [ Up ] [ Info ] [ Mail ]

Separation axioms. T_{1}-Space. Cofinite topology.
Hausdorff space. Regular and normal spaces.
Urysohn’s Lemma and Metrization Theorem.
Completely regular space. Tychonoff space.

Separation axioms

Def. T_{1}-Space. A T_{1}-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 T_{1}-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 T_{1}-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 T_{1}-space.

Theorem 1. A topological space X is a T_{1}-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 T_{1}-space if and only if τ contains the
cofinite topology on X.

T_{1}-topology. The cofinite topology on X is the coarsest
topology on X for which X with topology τ is a T_{1}-space .
Consequently the cofinite topology is also called the T_{1}-topology.

Def. Hausdorff space (or T_{2}-space). A Hausdorff space (or T_{2}-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 T_{1}-space.

Theorem 2. Every metric space is a Hausdorff space.

A T_{1}-space that is not Hausdorff. The topological space consisting of the real line R with the
cofinite topology, i.e. T_{1}-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 [T_{2}].

Thus we see that T_{1}-spaces need not be Hausdorff.

In general, a sequence {a_{1}, a_{2}, a_{3}, ...... }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. T_{3}-space. A regular T_{1}-space.

Def. T_{4}-space. A normal
T_{1}-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 T_{1} 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 = { f_{1}(x) = sin x, f_{2}(x) = sin 2x, f_{3}(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.

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 T_{1}-space.

Syn. T_{3 1/2}-space

● A Tychonoff space is a T_{3}-space and a T_{4}-space is a Tychonoff space.

Theorem 8. The class *C* [X, R] of all real-valued continuous functions on a completely
regular T_{1}-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

More from SolitaryRoad.com:

Jesus Christ and His Teachings

Way of enlightenment, wisdom, and understanding

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

On Self-sufficient Country Living, Homesteading

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

Theory on the Formation of Character

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies

Self-control, self-restraint, self-discipline basic to so much in life

[ Home ] [ Up ] [ Info ] [ Mail ]