tags:definition |topology

Topology

Context

Statements

Definition (@Tu11, Definition A.1). Let $X$ be a set, a family $\tau$ of subsets of $X$ is a topology on $X$ if

• The whole set and the empty set belong to the topology: $X\in \tau ,\varnothing \in \tau .$
• The union of any collection of subsets of $\tau$ belongs to $\tau$:

$$\{X_i{\}}_{i\in \mathbb{N}}\subseteq \tau \Rightarrow \bigcup _{i\in \mathbb{N}}{X}_i\in \tau .$$

• The intersection of any finite collection of subsets of $\tau$ belongs to $\tau$:

$$\{X_i{\}}_{i=1}^n\subseteq \tau \Rightarrow \bigcap _{i=1}^n{X}_i\in \tau .$$