Topological Preliminaries

Sets may be thick, thin and normal.

Topology is one of (quite a few) mathematical theories that permeate other branches of Mathematics connecting them into one coherent whole. I, too, have employed topological terminology on several occasions. However, as the example of reflection demonstrates, basing our intuitive perception of a topological transformation as an abstraction of a (physical) deformation might be questionable if not misleading. On this page, I wish to collect a few basic definitions and examples that might help acquaint you with the fundamentals of Topology. Most of the examples will be drawn on the 2-dimensional plane but, given the definitions of the distance and neighborhood could be carried over to the 1- and many dimensional cases.

There are many equivalent definitions of topology [Ref 2]. I pick the one based on the simplified notion of nearness. Following [Ref 1], I shall only consider circular neighborhoods which if applied consistently lead to exactly same definition of a topological space.

Definition

  1. A neighborhood of a point P∈R2 is a circular disk (or ball) D2(P,r) = {Q∈R2dist2(P, Q) < r}.

  2. Let S be a subset of the plane: S⊂R2. The point P is said to be near S if every neighborhood of P contains a point from S. If P is near S, we write P←S.

  3. A set S is called thick if every point of S is near S even after being removed from the set.

  4. A set S is called thin if none of its points satisfies that condition.

  5. The set of all points near S is called its closure: cl(S) = {P: P←S}.

  6. A set is closed if it coincides with its closure: S = cl(S). In other words, no point near S lies outside S.

  7. A point P∈S is internal to S iff if it's not near to Sc = R2 - S, the complement of S. Every internal point P∈S has a neighborhood U which is contained entirely in S: P∈U⊂S. The collection of all internal points of a set defines its interior.

  8. A set S is open if its complement Sc = R2 - S is closed which means none of its points is near its complement. Every point of an open set is internal to the set.

  9. A set is dense in R2 iff cl(S) = R2. Thick sets are also known as dense in themselves.

  10. A set S is dense in a set T iff S⊂T and T⊂cl(S).

  11. A set is nowhere dense if its closure has no internal points.

  12. A function (transformation) f: D→G is continuous iff for A⊂D and P←A, f(P)←f(A).

  13. A continuous function f that has a continuous inverse function is called a topological transformation, or homeomorphism.

  14. Two sets are said to be topologically equivalent if they are topological images of one another. Another term is homeomorphic.

  15. A point P near a set S is said to be its boundary point if it's also near its complement Sc. Thus, boundary (the set of all boundary points) is shared by a set and its complement. Moreover, boundary(S) = cl(S)∩cl(Sc).

Examples of sets

Below I'll use common notations Z, N, R, Q, C, for whole, integer (whole and positive), real, rational, and complex numbers. Also, let's agree that on a straight line dist2 coincides with the usual one - absolute value of the difference of two numbers. So that on a straight line, say, x-axis, the ball D(a, r) = {x: |a - x| < r} is just an open interval (a - r, a + r).

  1. The set Q of all rational numbers is dense in R, thick, neither open nor closed, and without internal points.

  2. The same is true for the set of all rational pairs Q×Q in R2 = R×R.

  3. The set Z of all whole numbers is thin, has no near points outside itself, closed, hence nowhere dense in R.

  4. A straight line is thick, closed and nowhere dense in R2.

  5. A neighborhood is open.

  6. Segment [a, b] = {x: a≤x≤b} is closed in R and dense in itself.

  7. Segment [a, b]×{0} = {(x, 0): a ≤ x ≤ b} is closed, thick and nowhere dense in R2 and dense in itself.

  8. Sequence xn = 1/n, where n > 0, outside itself has only one near point - the origin. It's thin and neither closed nor open. Augmented by 0, it becomes closed but remains nowhere dense. Its closure is neither thin nor thick.

Examples of transformations

Continuous transformations below are continuous because each transforms a ball into a ball

  1. Shift (or translation) f: D(0,r)→D(a,r), where f(x) = a + x is continuous and topological. For the vector addition it's also true in R2.

  2. Reflection in y-axis f: D((a, b), r)→D((-a, b), r), where f(x, y) = (-x, y) is continuous and topological.

  3. Stretching f: D(0, r)→D(0, ar), where f(x) = ax is continuous and topological.

  4. Folding f: R2→R2+, where R2+ is the right half-plane and f(x, y) = (|x|, y) is continuous but not topological.

  5. Shuffling f: [-1, 1]→[-1, 1], where f(x) = 1 + x if x ≤ 0, f(x) = -x if x > 0, is discontinuous but 1-1.

  6. Hilbert function f: [0, 1]→[0, 1]×[0, 1] is continuous but not topological.

There are nonhomeomorphic sets that are continuous 1-1 images of each other.

The notion of normality characterizes whole topological spaces rather than individual sets.

To define it, note that not only points may have neighborhoods. An open set is a neighborhood of any of its subsets. A topological space is normal iff any two closed non-intersecting subsets have non-intersecting neighborhoods.

Reference

  1. M. Henle, A Combinatorial Introduction to Topology, Dover Publications, NY, 1979
  2. K. Janich, Topology, UTM, Springer-Verlag, 1984
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