ACIT4330-Page/content/Lectures/Lecture 6 - 2.1 Topology.md

lecture, date

lecture	date
6	2025-01-27

Open Sets

The concept of Ball is required to understand what an 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 are unions of Ball.

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}