Lecture 6 - 2.1 Topology

Open Sets

The concept of balls is required to understand what an open sets are.

Definition

The definition can be found in Open Sets.
In other words ( ”):” ), for any open $\stackrel{\in x}{\overbrace{A}}\exists N$ such that $x_n\in A\forall n>N$.

In fact, open sets are unions of balls.

Topological Space

A Topological Space $X$ is a set with a Topology $\tau$.

Trivial Topology

$X$ is the set. $\tau =\{\varnothing ,X\}$.

Therefore $x\ne y$, then the only neighbour is $X$
$X\cap X=X$.

Discrete Topology

$X$ is the set. $\tau =\wp (X)$.

$\{x\}\in \tau$
$X\text{compact}⟺X\text{finite}$

This topological space is always Hausdorff as it includes all the points on $X$ are included

$A=\{x\},B=\{y\}A\cup B=\varnothing$

$d(x,y)=\{\begin{array}{ll}1,& \text{if}x\ne y\\ 0,& \text{if}x=y\end{array}$