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

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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 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 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 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 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 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 Y is continuous if and only if, for any p ε X and any A 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 Y is sequentially continuous at a point p ε X if and only if for every sequence {a_n} in X converging to p, the sequence {f(a_n)} in Y converges to f(p), i.e.

Sequential continuity and continuity are related as follows:

Theorem 4. If a function f : X 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 Y be a bijective (i.e. one-to-one and onto) mapping. If both the function f and the inverse function

f ^-1 : X 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).

References

1. Lipschutz. General Topology

2. Simmons. Introduction to Topology and Modern Analysis

3. Munkres. Topology, A First Course

4. James & James. Mathematics Dictionary