ACIT4330-Page/content/Definitions/Topological Spaces/Topology.md

Definition

A collection of subsets of X, called 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.

[!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}