# Definition
A collection of subsets of $X$, called [[Open Sets|open sets]], such that:
1. $X, \, \emptyset \in \tau$
2. Any union of sets from $\tau$ will be in $\tau$.
> [!info]-
> $$y, z \in \tau \implies y \cup z \in \tau$$
> $$x_{i} \in \tau \implies \cup_{i \in I} X_{i} \in \tau$$
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 [[Ball|balls]].

> [!example] Reasoning for having point/rule 3 in [[#Definition]]
> Consider the topology on $\mathbb{R}$ induced by the usual distance.
> $$B_{r}(x) = \langle x - r, x + r \rangle$$
> Note:
> $$\cap_{n \in \mathbb{N}} \langle -\frac{1}{n}, \frac{1}{n} \rangle = \{ 0\}$$
> ($\cap_{n \in \mathbb{N}}$ is an infinite intersection of all numbers (in $\mathbb{N}$))
> But the reason why this is not $= \{ 0, \varepsilon \}$ is a finite amount of intersections
> $\varepsilon \notin \langle -\frac{1}{n}, \frac{1}{n} \rangle$ for $n \gt \frac{1}{\varepsilon}$