Website owner:  James Miller

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

Continuous functions in topological spaces. Arbitrary closeness. Sequential continuity at a point. Open and closed mappings. Homeomorphism. Topological transformation.

Def. Continuous function (or mapping). A function f whose domain and range are topological spaces is continuous at a point x if for any neighborhood W of f(x) there is a neighborhood U of x such that W contains all points f(u) for which u is in U. Such a function is continuous if it is continuous at each point of its domain D. It then can be proved that f is continuous if and only if the inverse of each open set in the range R of f is open in D (or if and only if the inverse of each closed set in R is closed in D). For a function f whose domain and range are sets of real or complex numbers, this means f is continuous at x0 if f(x) can be made as nearly equal to f(x0) as one might wish by restricting x to be sufficiently close to x0; i.e. for any positive number ε there is a positive number δ such that |f(x) - f(x0)| < ε if |x - x0| < δ and x is in the domain of f.

A function is continuous on a set S if it is continuous at each point of S.


                                                                                                                                                James & James. Mathematics Dictionary

In mappings between metric spaces such as one, two, three or n-dimensional spaces the points sets in the above definition are sets of points in a continua (i.e. the continua of R, R2, R3 or Rn) and the open and closed sets are the usual open and closed sets in such continua. However, the points in the set X underlying a topological space X need not be points in a continuum. They can be discrete, unconnected points. Consider the following example.

Example 1. Let (X, D) be any discrete topological space and (Y, τ) be any topological space. Then every function f : X ole.gif Y is continuous. Why? Because if H is any open set of Y, its inverse image f-1[H] is an open set of X since every subset of a discrete space is open. The subsets of a discrete space are open due to the definition of a topological space (they are defined as open, called open).

We thus have continuous mappings defined on discrete sets of points.

Theorem 1. A function f : X ole1.gif Y is continuous if and only if the inverse of each member of a base B for Y is an open subset of X.

Theorem 2. Let β be a subbase for a topological space Y. A function f : X ole2.gif Y is continuous if and only if the inverse of each member of β is an open subset of X.

Arbitrary closeness. Let X be a topological space. A point p ε X is said to be arbitrarily close to a set A ole3.gif X if

1) p ε A



2) p is an accumulation point of A

From this definition we see that the closure of A


consists precisely of those points in X which are arbitrarily close to A. Also,


where ole6.gif is the interior of A and B(A) is the boundary of A. Thus a point p is arbitrarily close to A if it is either an interior point of A or a boundary point of A.

Theorem 3. A function f : X ole7.gif Y is continuous if and only if, for any p ε X and any A ole8.gif X,






Thus, from this theorem, we see that continuous functions can be characterized as those functions which preserve arbitrary closeness.

Sequential continuity at a point. A function f : X ole12.gif Y is sequentially continuous at a point p ε X if and only if for every sequence {an} in X converging to p, the sequence {f(an)} in Y converges to f(p), i.e.



Sequential continuity and continuity are related as follows:

Theorem 4. If a function f : X ole14.gif Y is continuous at p ε X, then it is sequentially continuous at p.

Note. The converse of the above theorem is not necessarily true.



Open and closed mappings. A mapping (correspondence, transformation, or function) which associates with each point of a space D a unique point of a space Y, is open if the image of each open set of D is open in the range R; it is closed if the image of each closed set is closed. An open mapping might or might not be closed and a closed mapping might or might not be open. Also a continuous mapping need not be either open or closed.

Def. Homeomorphism. Let X and Y be topological spaces. Let f : X ole15.gif Y be a bijective (i.e. one-to-one and onto) mapping. If both the function f and the inverse function

            f -1 : X ole16.gif Y

are continuous, then f is called a homeomorphism.

Thus a homeomorphism is a bicontinuous, bijective mapping of one topological space onto another. A homeomorphism can also be described as a bijective mapping of one topological space onto another that is both continuous and open since a mapping that is both continuous and open is bicontinuous.

A homeomorphism is also called a topological transformation.

What is the significance of a homeomorphism or topological transformation?

Def. Topological transformation. A one-to-one correspondence between the points of two geometric figures A and B which is continuous in both directions; a one-to-one correspondence between the points of A and B such that open sets in A correspond to open sets in B, and conversely (or analogously for closed sets). If one figure can be transformed into another by a topological transformation, the two figures are said to be topologically equivalent. Continuous deformations are examples of topological transformations.

Syn. Homeomorphism

                                                                                                            James & James. Mathematics Dictionary.

Topological property. Any property of a geometrical figure A that holds as well for every figure into which A may be transformed by a topological transformation. Examples are the properties of connectedness and compactness, of subsets being open or closed, and of points being limit points.

                                                                                                James & James. Mathematics Dictionary.

Topology is primarily concerned with the investigation of properties that are invariant under homeomorphisms (or topological transformations).


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

The Way of Truth and Life

God's message to the world

Jesus Christ and His Teachings

Words of Wisdom

Way of enlightenment, wisdom, and understanding

Way of true Christianity

America, a corrupt, depraved, shameless country

On integrity and the lack of it

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

Who will go to heaven?

The superior person

On faith and works

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

The teaching is:

On modern intellectualism

On Homosexuality

On Self-sufficient Country Living, Homesteading

Principles for Living Life

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

America has lost her way

The really big sins

Theory on the Formation of Character

Moral Perversion

You are what you eat

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

Cause of Character Traits --- According to Aristotle

These things go together


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

The True Christian

What is true Christianity?

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Walking a solitary road

Intellectual disparities among people and the power in good habits

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

On responding to wrongs

Real Christian Faith

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

Sizing up people

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

We are our habits

What creates moral character?

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