The definition of a topology on a set

A topology on a set \(X\) is a collection \(\tau\subseteq\mathcal P(X)\) such that:

1. \(\varnothing\in\tau\) and \(X\in\tau\);
2. if \(\{U_\alpha\}_{\alpha\in A}\subseteq\tau\), then \(\bigcup_{\alpha\in A}U_\alpha\in\tau\);
3. if \(U_1,\dots,U_n\in\tau\), then \(U_1\cap\cdots\cap U_n\in\tau\).

The members of \(\tau\) are called the open subsets of \(X\), and the pair \((X,\tau)\) is a topological space.