Topology/Lesson 1

What is a Topology?
The word "topology" has two meanings: it is both the name of a mathematical subject and the name of a mathematical structure. A topology on a set $$X$$ (as a mathematical strucure) is a collection of what are called "open subsets" of $$X$$ satisfying certain relations about their intersections, unions and complements. In the basic sense, Topology (the subject) is the study of structures arising from or related to topologies.

Reading Assignment

 * The following reading is suggested to help supplement this lesson.
 * Wikibooks Topology book:
 * Chapter 2.2.2
 * Chapter 2.2.3
 * Chapter 2.2.4
 * Wikipedia articles:
 * Topology
 * General topology
 * Open set

Definition (topology)
Let $$X$$ be a set. Then a topology on $$X$$ is a set $$\mathcal T$$ such that the following conditions hold. The set $$X$$ together with the topology $$\mathcal T$$ is called a topological space (or simply a space) and is commonly written as the pair $$(X,\mathcal T).$$ Or, when $$\mathcal T$$ is understood it may be omitted and we will simply say that $$X$$ is a topological space.
 * 1) $$\{\emptyset, X\}\subset \mathcal T\subset 2^X$$ (where $$2^X$$ denotes the power set of X)
 * 2) For $$\mathcal S\subset \mathcal T$$ we have $$\left(\bigcup_{U\in \mathcal S}U\right)\in \mathcal T.$$
 * 3) For finite sets $$\mathcal S\subset \mathcal T$$ we have $$\left(\bigcap_{U\in \mathcal S}U\right)\in \mathcal T.$$

Examples
Here are some very simple examples of topological spaces. For these examples, $$X$$ can be any set.
 * Discrete topology
 * The collection $$\mathcal T_d=2^X$$ is called the discrete topology on $$X.$$


 * Indiscrete topology
 * The collection $$\mathcal T_i=\{\emptyset,X\}$$ is called the indiscrete topology or trivial topology on $$X.$$


 * Particular point topology
 * Given a point $$x_0\in X,$$ the collection $$\mathcal T_{x_0}=\{U\subset X\mid x_0\in U\}\cup \{\emptyset\}$$ is called the particular-point topology on $$X.$$

It is left as an exercise to verify that each of these three collections does indeed satisfy the axioms of a topology (conditions 1,2,3 in the definition above).

Reading supplement
See also Wikipedia articles:
 * Discrete space
 * Trivial topology
 * Particular point topology

Definition (open set, closed set,neighborhood)
Suppose that $$(X,\mathcal T)$$ is a topological space.
 * Open set
 * A set $$U\subset X$$ is open if $$U\in \mathcal T.$$


 * Closed set
 * A set $$A\subset X$$ is closed if $$A^c=(X\setminus A)\in \mathcal T.$$


 * Neighborhood
 * For a point $$x_0\in X$$ a set $$N\subset X$$ is a neighborhood of $$x_0$$ if there is an open set $$U\in \mathcal T$$ such that $$x_0\in U\subset N.$$

Definition (closed topology)
Suppose that $$\{\emptyset,X\}\subset \mathcal S\subset 2^X.$$ Then $$\mathcal S$$ is a closed topology if Show that for any set $$X,$$ the collection $$\mathcal T$$ is a topology on $$X$$ if and only if the collection $$\mathcal S=\{T^c\mid T\in \mathcal T\}$$ is a closed topology on $$X.$$
 * Alternate definition of a topology
 * 1) for any $$\mathcal R\subset \mathcal S$$ we have $$\left(\bigcap_{R\in \mathcal R}R\right)\in \mathcal S$$ and
 * 2) for any finite collection $$\mathcal R\subset \mathcal S$$ we have $$\left(\bigcup_{R\in \mathcal R}R\right)\in \mathcal S.$$

Definition (interior, closure)
Let $$(X,\mathcal T)$$ be a space and let $$A\subset X.$$
 * Interior
 * The interior of $$A$$ (denoted $$\operatorname{int}(A)$$) is defined to be the union of all open sets contained in $$A.$$ In other words, $$\operatorname{int}(A)=\bigcup_{\underset{U\in \mathcal T}{U\subset A}}U.$$


 * Closure
 * The closure of $$A$$ (denoted $$\bar A$$) is defined to be the intersection of all closed sets containing $$A.$$ That is, $$\bar A = \bigcap_{\underset{B^c\in \mathcal T}{B\supset A}}B.$$

Definition (basis)
Let $$(X,\mathcal T)$$ be a space. Then a collection $$\mathcal B\subset \mathcal T$$ is a basis if for any point $$x_0\in X$$ and any neighborhood $$N$$ of $$x_0$$ there is a basis element $$B\in \mathcal B$$ such that $$x_0\in B\subset N.$$

The benefit of talking about a basis is that sometimes describing every open set is unwieldy. For example, describing an open set in the Euclidean plane $$\mathbb{R}^2$$ would be difficult, but describing a basis is very easy. A basis of open sets in the plane is given by "open rectangles". That is $$\mathcal B=\{(a,b)\times (c,d)\mid a<b, c<d\in \mathbb{R}\}$$ forms a basis.

Once a basis is determined, a set $$U\subset X$$ is open if it is the union of basis elements. That is, if $$\mathcal B$$ is a basis, then the topology is given by $$\mathcal T=\left\{\bigcup_{B\in \mathcal A}B \mid \mathcal A\subset \mathcal B\right\}.$$

Definition (compact)
Let $$(X,\mathcal T)$$ be a topological space. Then a set $$K\subset X$$ is compact if and only if every open cover of $$K$$ has a finite subcover.

Reading supplement
See also Wikipedia articles:
 * Interior
 * Closure
 * Basis

Lesson Exercises

 * 1) Let $$X$$ be a three-point set. Then there are $$2^{2^3}=256$$ different subsets of $$2^X.$$  How many of these are topologies on $$X?$$  In other words, how many different 3-point topologies are there?
 * 2) Can you find a formula for the number of topologies on an $$n$$-point set?
 * 3) Suppose that $$\mathcal B\subset 2^X$$ is such that for any $$x\in X$$ there is a set $$B\in \mathcal B$$ containing $$x$$ and that for any two sets $$B_1,B_2\in \mathcal B$$ such that $$B_1\cap B_2\ne \emptyset$$ there is a set $$B_3\in \mathcal B$$ such that $$B_3\subset B_1\cap B_2.$$ Show that the collection $$\mathcal T=\left\{\bigcup_{A\in\mathcal A}A\mid \mathcal A\subset \mathcal B\right\}\cup \{\emptyset\}$$ is a topology on $$X$$ and that $$\mathcal B$$ is a basis for $$\mathcal T.$$
 * 4) Let $$\mathcal S\subset 2^X$$ be such that for all $$x\in X$$ there is a set $$S\in \mathcal S$$ which contains $$x.$$ Then show that the collection $$\mathcal B=\left\{\bigcap_{R\in \mathcal R}R\mid \mathcal R\subset \mathcal S\text{ is finite}\right\}$$ is a basis for a topology $$\mathcal  T$$ on $$X$$ (using the criterion given in exercise 3).  In this case, we call $$\mathcal S$$ a subbasis for $$\mathcal T.$$
 * 5) A basis $$\mathcal B$$ for a topology $$\mathcal T$$ is said to be minimal if any proper collection $$\mathcal A\subsetneq \mathcal B$$ is not a basis for $$\mathcal T.$$ Given a set $$X,$$ find a minimal basis for the discrete topology $$\mathcal T_d=2^X.$$
 * 6) It is clear from the definition that $$\operatorname{int}(A)\subset A\subset \bar A.$$ Show that if $$A\subset B$$ then $$\operatorname{int}(A)\subset\operatorname{int}(B)$$ and $$\bar A\subset \bar B.$$
 * 7) Show that $$\operatorname{int}(\operatorname{int}(A))=\operatorname{int}(A)$$ and that $$\overline{\bar A}=\bar A.$$ Use these facts to show that $$A\subset X$$ is open if and only if $$A=\operatorname{int}(A)$$ and is closed if and only if $$A=\bar A.$$
 * 8) Is it true that for any set $$A\subset X$$ that $$\overline{A^c}=(\operatorname{int}(A))^c?$$ Give a proof or a counterexample.
 * 9) Show that the collection $$\mathcal B=\{(a,b)\mid a<b\in \mathbb{R}\}$$ of open intervals is a basis for a topology on $$\mathbb{R}.$$ This is called the standard topology on $$\mathbb{R}.$$

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