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 ${\displaystyle X}$ (as a mathematical strucure) is a collection of what are called "open subsets" of ${\displaystyle X}$ 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.

The following reading is suggested to help supplement this lesson.
• Wikibooks Topology book:
• Wikipedia articles:

Definition (topology)

Let ${\displaystyle X}$ be a set. Then a topology on ${\displaystyle X}$ is a set ${\displaystyle {\mathcal {T}}}$ such that the following conditions hold.

1. ${\displaystyle \{\emptyset ,X\}\subset {\mathcal {T}}\subset 2^{X}}$ (where ${\displaystyle 2^{X}}$ denotes the power set of X)
2. For ${\displaystyle {\mathcal {S}}\subset {\mathcal {T}}}$ we have ${\displaystyle \left(\bigcup _{U\in {\mathcal {S}}}U\right)\in {\mathcal {T}}.}$
3. For finite sets ${\displaystyle {\mathcal {S}}\subset {\mathcal {T}}}$ we have ${\displaystyle \left(\bigcap _{U\in {\mathcal {S}}}U\right)\in {\mathcal {T}}.}$

The set ${\displaystyle X}$ together with the topology ${\displaystyle {\mathcal {T}}}$ is called a topological space (or simply a space) and is commonly written as the pair ${\displaystyle (X,{\mathcal {T}}).}$ Or, when ${\displaystyle {\mathcal {T}}}$ is understood it may be omitted and we will simply say that ${\displaystyle X}$ is a topological space.

Examples

Here are some very simple examples of topological spaces. For these examples, ${\displaystyle X}$ can be any set.

Discrete topology
The collection ${\displaystyle {\mathcal {T}}_{d}=2^{X}}$ is called the discrete topology on ${\displaystyle X.}$
Indiscrete topology
The collection ${\displaystyle {\mathcal {T}}_{i}=\{\emptyset ,X\}}$ is called the indiscrete topology or trivial topology on ${\displaystyle X.}$
Particular point topology
Given a point ${\displaystyle x_{0}\in X,}$ the collection ${\displaystyle {\mathcal {T}}_{x_{0}}=\{U\subset X\mid x_{0}\in U\}\cup \{\emptyset \}}$ is called the particular-point topology on ${\displaystyle X.}$

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).

Definition (open set, closed set,neighborhood)

Suppose that ${\displaystyle (X,{\mathcal {T}})}$ is a topological space.

Open set
A set ${\displaystyle U\subset X}$ is open if ${\displaystyle U\in {\mathcal {T}}.}$
Closed set
A set ${\displaystyle A\subset X}$ is closed if ${\displaystyle A^{c}=(X\setminus A)\in {\mathcal {T}}.}$
Neighborhood
For a point ${\displaystyle x_{0}\in X}$ a set ${\displaystyle N\subset X}$ is a neighborhood of ${\displaystyle x_{0}}$ if there is an open set ${\displaystyle U\in {\mathcal {T}}}$ such that ${\displaystyle x_{0}\in U\subset N.}$

Definition (closed topology)

Alternate definition of a topology

Suppose that ${\displaystyle \{\emptyset ,X\}\subset {\mathcal {S}}\subset 2^{X}.}$ Then ${\displaystyle {\mathcal {S}}}$ is a closed topology if

1. for any ${\displaystyle {\mathcal {R}}\subset {\mathcal {S}}}$ we have ${\displaystyle \left(\bigcap _{R\in {\mathcal {R}}}R\right)\in {\mathcal {S}}}$ and
2. for any finite collection ${\displaystyle {\mathcal {R}}\subset {\mathcal {S}}}$ we have ${\displaystyle \left(\bigcup _{R\in {\mathcal {R}}}R\right)\in {\mathcal {S}}.}$

Show that for any set ${\displaystyle X,}$ the collection ${\displaystyle {\mathcal {T}}}$ is a topology on ${\displaystyle X}$ if and only if the collection ${\displaystyle {\mathcal {S}}=\{T^{c}\mid T\in {\mathcal {T}}\}}$ is a closed topology on ${\displaystyle X.}$

Definition (interior, closure)

Let ${\displaystyle (X,{\mathcal {T}})}$ be a space and let ${\displaystyle A\subset X.}$

Interior
The interior of ${\displaystyle A}$ (denoted ${\displaystyle \operatorname {int} (A)}$) is defined to be the union of all open sets contained in ${\displaystyle A.}$ In other words, ${\displaystyle \operatorname {int} (A)=\bigcup _{\underset {U\in {\mathcal {T}}}{U\subset A}}U.}$
Closure
The closure of ${\displaystyle A}$ (denoted ${\displaystyle {\bar {A}}}$) is defined to be the intersection of all closed sets containing ${\displaystyle A.}$ That is, ${\displaystyle {\bar {A}}=\bigcap _{\underset {B^{c}\in {\mathcal {T}}}{B\supset A}}B.}$

Definition (basis)

Let ${\displaystyle (X,{\mathcal {T}})}$ be a space. Then a collection ${\displaystyle {\mathcal {B}}\subset {\mathcal {T}}}$ is a basis if for any point ${\displaystyle x_{0}\in X}$ and any neighborhood ${\displaystyle N}$ of ${\displaystyle x_{0}}$ there is a basis element ${\displaystyle B\in {\mathcal {B}}}$ such that ${\displaystyle x_{0}\in B\subset N.}$

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 ${\displaystyle \mathbb {R} ^{2}}$ 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 ${\displaystyle {\mathcal {B}}=\{(a,b)\times (c,d)\mid a forms a basis.

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

Definition (compact)

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

Lesson Exercises

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

Notes

1. where ${\displaystyle (a,b):=\{x\in \mathbb {R} :a