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

### 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)
#### Alternate definition of a topology

Suppose that $\{\emptyset ,X\}\subset {\mathcal {S}}\subset 2^{X}.$ Then ${\mathcal {S}}$ is a closed topology if
* for any ${\mathcal {R}}\subset {\mathcal {S}}$ we have $\left(\bigcap _{R\in {\mathcal {R}}}R\right)\in {\mathcal {S}}$ and
* for any finite collection ${\mathcal {R}}\subset {\mathcal {S}}$ we have $\left(\bigcup _{R\in {\mathcal {R}}}R\right)\in {\mathcal {S}}.$
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 $X.$

### 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

* 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?
* Can you find a formula for the number of topologies on an $n$-point set?
* 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 $B_{1}\cap B_{2}\neq \emptyset$ 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 ${\mathcal {T}}.$
* 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 ${\mathcal {T}}.$
* 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 ${\mathcal {T}}_{d}=2^{X}.$
* 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 ${\bar {A}}\subset {\bar {B}}.$
* 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 $A={\bar {A}}.$
* Is it true that for any set $A\subset X$ that ${\overline {A^{c}}}=(\operatorname {int} (A))^{c}?$ Give a proof or a counterexample.
* 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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