A collection \(\tau\) of subsets of a set \(X\) is called topology if it satisfies
- \(\emptyset, X \in \tau\),
- \(U_i \in \tau\) for \(i \in I\), then \(\bigcup_{i \in I} U_i \in \tau\),
- \(U, V \in \tau\), then \(U\cap V \in \tau\).
The sets of \(\tau\) are called open and the tuple \((X,\tau)\) topological space.
Remarks
- The topology is a notion of nearness. We call subset \(U\in \tau\) neighbourhood of some point \(x\in X\) if \(x\in U\).
- By induction the 3rd condition implies \(U_i \in \tau\) for \(i \in \{1,\ldots,n\}\), then \(\bigcap_{i \in I} U_i \in \tau\).
Examples