---
lecture: 6
date: 2025-01-27
---
# Open Sets
The concept of [[Ball|balls]] is required to understand what an [[Open Sets|open sets]] are.
## Definition
The definition can be found in [[Open Sets]].
In other words ( "):" ), for any open $\overbrace{A}^{\in x} \exists N$ such that $x_{n} \in A \; \forall n \gt N$.

In fact, [[Open Sets|open sets]] are unions of [[Ball|balls]].
# Topological Space
A [[Topological Space]] $X$ is a set with a [[Topology]] $\tau$.

> [!example] Trivial [[Topology]]
> $X$ is the set. $\tau = \{ \emptyset, X \}$.
> 
> Therefore $x \neq y$, then the only neighbour is $X$
> $X \cap X = X$.

> [!example] Discrete [[Topology]]
> $X$ is the set. $\tau = \wp(X)$.
> 
> $\{ x \} \in \tau$
> $X \, \text{compact} \iff X \, \text{finite}$
> 
> This topological space is always [[Hausdorff]] as it includes all the points on $X$ are included
> > [!note]
> > $$A = \{ x \}, \; B = \{ y \} \; A \cup B = \emptyset$$
> 
> $$d(x, y) = \begin{cases} 1, & \text{if}\ x \neq y\\ 0, & \text{if} \, x = y\end{cases}$$