Topology

Definition

A collection of subsets of $X$, called open sets, such that:

1. $X,\varnothing \in \tau$
2. Any union of sets from $\tau$ will be in $\tau$.

Info

$y,z\in \tau ⟹y\cup z\in \tau$
$x_i\in \tau ⟹{\cup }_{i\in I}{X}_i\in \tau$

3. Any finite intersection of sets from $\tau$ will be in $\tau$.

Info

$y\cap z\in \tau$
${\cap }_{i\in I}{X}_i\notin \tau$

Examples

Example

The topology induced by a metric on $X$ is the collection of all unions of balls.

Reasoning for having point/rule 3 in Definition

Consider the topology on $\mathbb{R}$ induced by the usual distance.
$B_r(x)=⟨x-r,x+r⟩$
Note:
${\cap }_{n\in \mathbb{N}}⟨-\frac{1}{n},\frac{1}{n}⟩=\{0\}$
(${\cap }_{n\in \mathbb{N}}$ is an infinite intersection of all numbers (in $\mathbb{N}$))
But the reason why this is not $=\{0,\epsilon \}$ is a finite amount of intersections
$\epsilon \notin ⟨-\frac{1}{n},\frac{1}{n}⟩$ for $n>\frac{1}{\epsilon }$