# topological space

## 1. definition

\(\mathcal{T} \subseteq X\) is called a topology of \(X\) if:

- \(\emptyset, X \in \mathcal{T}\)
- \(A,B\in \mathcal{T} \Rightarrow A\cap B \in \mathcal{T}\)
- For index set \(\mathcal{I}\), with \((A_{i})_{i\in \mathcal{I}}\), \(\bigcup_{i\in\mathcal{I}} A_i \in \mathcal{T}\)
- note that a topology is closed under
*finite*intersections and infinite unions

## 2. related

- superficially similar to Sigma Field – a measurable space. But there are key differences see here
- youtube video