# 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

A

*neighborhood*of a point P∈R^{2}is a circular disk (or*ball*) D_{2}(P,r) = {Q∈R^{2}: dist_{2}(P, Q) < r}.Let S be a subset of the plane: S⊂R

^{2}. 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 writeP←S. A set S is called

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

*thin*if none of its points satisfies that condition.The set of all points near S is called its

*closure*:cl(S) = {P: P←S}. A set is

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

*internal*to S iff if it's not near toS 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^{c}= R^{2}- S,*interior*.A set S is

*open*if its complement S^{c}= R^{2}- S is closed which means none of its points is near its complement. Every point of an open set is internal to the set.A set is

*dense*in R^{2}iffcl(S) = R Thick sets are also known as^{2}.*dense in themselves.*A set S is

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

*nowhere dense*if its closure has no internal points.A function (

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

*inverse*function is called a*topological transformation*, or*homeomorphism*.Two sets are said to be

*topologically equivalent*if they are topological images of one another. Another term is*homeomorphic*.A point P near a set S is said to be its

*boundary point*if it's also near its complement S^{c}. Thus,*boundary*(the set of all boundary points) is shared by a set and its complement. Moreover,boundary(S) = cl(S)∩cl(S ^{c}).

## 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 dist_{2} coincides with the usual one - absolute value of the difference of two numbers. So that on a straight line, say, x-axis, the ball

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

The same is true for the set of all rational pairs Q×Q in R

^{2}= R×R.The set Z of all whole numbers is thin, has no near points outside itself, closed, hence nowhere dense in R.

A straight line is thick, closed and nowhere dense in R

^{2}.A neighborhood is open.

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

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

^{2}and dense in itself.Sequence x

_{n}= 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

*Shift*(or*translation*) f: D(0,r)→D(a,r), wheref(x) = a + x is continuous and topological. For the*vector addition*it's also true in R^{2}.*Reflection*in y-axis f: D((a, b), r)→D((-a, b), r), wheref(x, y) = (-x, y) is continuous and topological.*Stretching*f: D(0, r)→D(0, ar), wheref(x) = ax is continuous and topological.*Folding*f: R^{2}→R^{2+}, where R^{2+}is the right half-plane andf(x, y) = (|x|, y) is continuous but not topological.*Shuffling*f: [-1, 1]→[-1, 1], wheref(x) = 1 + x ifx ≤ 0, f(x) = -x ifx > 0, is discontinuous but 1-1.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

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

### Absolute Value

- Absolute Value
- Importance of the Absolute Value
- Cartesian Coordinate System
- Algebraic Structure of Complex Numbers
- Structure of Hyperreal Numbers
- Farmer and Wife To Catch Rooster and Hen
- Topological Preliminaries
- Proizvolov's Identity
- Absolute Value and 1998 integers
- Optimal Residence
- Sum of Absolute Values

### Elements of Topology

- Topological Preliminaries
- When Index Equals Content: Sperner Lemma
- Regular Polyhedra. Euler Characteristic
- Tarski-Banach Theorem
- Cantor set and function
- Shaggy Dog Theorem
- Knot Theory
- Crossing Number of a Graph

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