Topological Space

Part of Topology.

Definition

A Topological Space is a pair $(X,\tau )$ where $X$ is a set and $\tau$ is a collection of subsets of X called open sets.

Axioms

1. Empty set and whole space are open:

$$\varnothing \in \tau \text{and}X\in \tau$$

2. Arbitrary unions of open sets are open: if $\{U_a{\}}_{a\in I}$ is any collection of sets in $\tau$, then

$$\bigcup _{a\in I}{U}_a\in \tau$$

3. Finite intersections of open sets are open: if $U_1,U_2,\dots ,U_n\in \tau$ then

$$U_1\cap U_2\cap \cdots \cap U_n\in \tau$$