Notation

$X$ : a set
$P(X)$ : the power set of $X$ , that is, the set of all subsets of $X$ .
$\mathbb{R}$ : the set of real numbers.

Topological spaces

Topology

A topology on $X$ is a subset $\tau\subseteq P(X)$ , such that:
- (T1): $\emptyset,X\in\tau$ .
- (T2): If $U_{\alpha}\in\tau$ for all $\alpha\in I$ , then $\cup_{\alpha}U_{\alpha}\in\tau$ .
- (T3): If $U_{1},U_{2}\in\tau$ , then $U_{1}\cap U_{2}\in\tau$ .
Topological spaces
1. If $\tau$ is a topology on $X$ , the pair $(X,\tau)$ is called a topological space.
2. The elements of $\tau$ are called open sets.

Let $X$ be any set. Then $\tau_{d}=P(X)$ is a topology, called the discrete topology.
Let $X$ be any set. Then $\tau_{i}=\{\emptyset,X\}$ is a topology, called the indiscrete topology.
Let $X=\{1,2\}$ . Then $\tau=\{\emptyset,X,\{1\}\}$ is a topology, and $(X,\tau)$ is called the Sierpiński space.

Let
- $X=\mathbb{R}$ ,
- $\tau=\{U\subseteq X\mid \forall x\in U\,\exists\epsilon>0, \text{such that } (x-\epsilon,x+\epsilon)\subseteq U\}$ .
- Then $\tau$ is called the usual topology on $\mathbb{R}$ .
Let
- $(X,d)$ be any metric space. For $x\in X$ , let $B_{\epsilon}(x)=\{y\in X\mid d(x,y)<\epsilon\}$ .
- $\tau=\{U\subseteq X\mid \forall x\in U\,\exists\epsilon>0, \text{such that } B_{\epsilon}(x)\subseteq U\}$ .
- Then $\tau$ is called the metric topology on $X$ .

Cofinite topology

Let $\tau=\{U\subseteq X\mid X-U \text{ is finite}\}\cup\{\emptyset\}$ . Then $\tau$ is a topology on $X$ .
Sketch of proof
- $\emptyset\in\tau$ immediately. $X\in\tau$ because $X-X=\emptyset$ is finite.
- Let $U_{\alpha}\in\tau$ for all $\alpha\in I$ . We need to show that either $\cup_{\alpha}U_{\alpha}$ is empty or $X-\cup_{\alpha}U_{\alpha}$ is finite.
- Let $U_{1},U_{2}\in\tau$ . We need to show that either $U_{1}\cap U_{2}$ is empty or $X-(U_{1}\cap U_{2})$ is finite.

Show that if $X$ is any set, there is a metric on $X$ such that the metric topology is the same as the discrete topology.
Show that in general there is no metric on $X$ such that the metric topology is the same as the indiscrete topology.
Show that if $\tau$ is any topology on $X$ , then $\tau_{i}\subseteq\tau\subseteq\tau_{d}$ .
Show that if $\tau_{\alpha}$ is a topology on $X$ for each $\alpha\in I$ , then $\tau=\cap\tau_{\alpha}$ is a topology on $X$ .
Give an example of a set $X$ and topologies $\tau_{1},\tau_{2}$ such that $\tau_{1}\cup\tau_{2}$ is not a topology on $X$ .
Let $\tau=\{(a,\infty)\mid a\in \mathbb{R}\}\cup\{\mathbb{R},\emptyset\}$ . Prove that $\tau$ is a topology on $\mathbb{R}$ .
Let $a\in X$ , and $\tau_{a}=\{U\subseteq X\mid a\in U\}\cup\{\emptyset\}$ . Prove that $\tau_{a}$ is a topology on $X$ .
Let $a\in X$ , and $\rho_{a}=\{U\subseteq X\mid a\not\in U\}\cup\{X\}$ . Is $\rho_{a}$ a topology on $X$ ?
Let $X=\{1,2,3\}$ . Enumerate all topologies on $X$ .