SolitaryRoad.com

Website owner: James Miller

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

Bases, subbases for a topology. Subspaces. Relative topologies.

Def. Base for a topology. Let (X, τ) be a topological space. A class B of open sets is a base for the topology of X if each open set of X is the union of some of the members of B.

Syn. Basis

Example 1. The open intervals on the real line form a base for the collection of all open sets of real numbers i.e. the usual topology on R.

Example 2. The open discs in the plane
form a base for the collection of all open
sets in the plane R^{2} i.e. the usual
topology on R^{2}. The open rectangles in
the plane also form a base for the
collection of all open sets in the plane.

Example 3. Let X represent the open point set shown in Fig. 1 with a topology τ consisting of all open sets in X. The open spheres in space form a base for topology τ. The open rectangular parallelepipeds in space also form a base for τ.

Example 4. Let X be any discrete space with topology D. Then the collection

B = {{p}: p ε X}

of all singleton subsets of X is a base for the discrete topology D.

What conditions must a collection of subsets meet in order to be a base for some topology of a set X? The answer is given by the following theorem:

Theorem 1. Let B be a collection of subsets of a set X. Then B is a base for some topology on X if and only if it possesses the following two properties:

1) X = {B: B ε B}

2) For any B, B* ε B, B B* is the union of members of B,

or, equivalently,

if p ε B
B*, then there exist a B_{p} ε B such
that p ε B_{p} where B_{p }is a subset of B
B*.

Def. Subbase for a topology. Let (X, τ) be a topological space. A class S of open sets is a subbase for the topology τ on X if the collection of all finite intersections of members of S is a base for the topology.

Example 5. Every open interval (a, b) in the real line R is the intersection of two infinite open intervals (a, ) and (- , b) i.e. (a, b) = (a, ) (- , b). The open intervals form a base for the usual topology on R and the collection of all of these infinite open intervals is a subbase for the usual topology on R.

Example 6. The intersection of a vertical and a horizontal infinite open strip in the plane is an
open rectangle as shown in Fig. 2. The open rectangles form a base for the usual topology on R^{2}
and the collection of all infinite open strips (horizontal and vertical) is a subbase for the usual
topology on R^{2}.

Topologies generated by collections of sets. Let A be any class of sets of a set X. Although A may not be a base for a topology on X it always generates a topology on X in the following sense:

Theorem 2. Any class A of subsets of a non-empty set X is the subbase for a unique topology T on X. That is, finite intersections of members of A form a base for the topology T on X.

Theorem 3. Let A be a class of subsets of a non-empty set X. Then the topology T on X generated by A is the intersection of all topologies on X which contain A.

Base for the neighborhood system of a point p (or a local base at p). Let p be a point in a topological space X. A collection N of open sets is a base for the neighborhood system of a point p (or a local base at p) if p belongs to each member of N and any open set which contains p also contains a member of N.

Example 7. Consider the collection of all open sets in the plane R^{2} i.e. the usual topology on R^{2}.
Then the collection B_{p} of all open discs centered at p is a local base at p because any open set K
that contains p also contains an open disc D_{p} whose center is p. See Fig. 3.

Example 8. Consider the collection of all open sets of real numbers i.e. the usual topology on R. The collection of all open intervals (a - δ, a + δ) with center at a is a local base at point a.

Example 9. Let X be any discrete space and let p ε X. Then a local base at point p is the singleton set {p}.

Theorem 4. Let B be a base for a topology T on a topological space X and let p ε X. Then the members of the base B which contain p form a local base at the point p.

Theorem 5. A point p in a topological space X is a limit point of a subset A of X if and only if
each member of some local base B_{p} at p contains a point of A different from p.

Theorem 6. A sequence {a_{1}, a_{2}, ..... } of points in a topological space X converges to p ε X if
and only if each member of some local base B_{p} at p contains almost all, i.e. all but a finite number,
of the terms of the sequence.

Subbase for the neighborhood system of a point p (or a local subbase at p). Let p be a point in a topological space X. A subbase for the neighborhood system of a point p (or a local subbase at p) is a collection S of sets such that the collection of all finite intersections of members of S is a base for the neighborhood system of p.

****************************************************************************

Subspaces, relative topologies. Let (X, T) be a topological space. Let A be a subset of X.
Then the collection T_{A} of all intersections of A with the open sets of T is a topology on A, called
the subspace topology or relative topology on A. The topological space A with topology T_{A} is
called a subspace of X. The members of T_{A} are open sets in the sense of the definition of a
topological space. They are called open because they form a topology but may not be the same
open sets as those of T.

Example 4. Let X be the real line R with the usual topology, the set of all open sets on the real
line. An open set on the real line is some collection of open intervals such as that shown in Fig. 4.
Let A be some interval [a, b] of the real line. Then the relative topology on [a, b] is the collection
T_{A} of all intersections of [a, b] with the set of all open sets of R. The open sets of T_{A} will consist
of closed /open sets of type [a, b) and (a, b]. The topological space A with topology T_{A} is a
subspace of R.

Example 5. Let X be the plane R^{2} with the usual topology, the set of all open sets in the plane.
An open set in R^{2} is a set such as that shown in Fig. 5. Let A be the rectangular region in R^{2} given
by

x_{1}
x
x_{2}

y_{1}
y
y_{2}

Then the relative topology on A is the
collection T_{A} of all intersections of A with the
set of all open sets of R^{2}. The open sets of T_{A}
consist of partially open / partially closed sets.
The topological space A with topology T_{A} is a
subspace of R^{2}.

When dealing with a space X and a subspace
A, one must be careful in using the term “open
set”. Does he mean an open set of T or of T_{A}?
If A is a subspace of X, we say that a set U is
open in A (or open relative to A) if it belongs
to T_{A}. We say that U is open in X if it belongs to T.

There is a special situation in which every set open in A is also open in X:

Theorem 7. Let A be a subspace of X. If a set U is open in A and A is open in X, then U is open in X.

Theorem 8. Let A be a subset of X. If B is a base for the topology of X, then the collection

B_{A} = {B
A: B ε B}

is a base for the subspace topology on A.

Example 6. Let X be the real line R with the usual topology. Let A = [a, b] be a subset of X. A
base B for the usual topology on R is the set of all open intervals (a, b). A base B_{A} for the
subspace topology on A is the collection of all intersections of [a, b] with the set of all open
intervals (a, b) i.e. a collection of closed /open sets of type [a, b) and (a, b].

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 ]