Topological space

A topological space is a set $X$ together with a topology $\tau$, that is, a collection of subsets of $X$, called the open sets, satisfying the following axioms:

1. The empty set and the whole space are open: $\emptyset\in \tau$ and $X\in \tau$.
2. Arbitrary unions of open sets are open: if $\{U_{i}\}_{i\in I}$ is any family of sets in $\tau$, then $\bigcup_{i \in I} U_i \in \tau$.
3. Finite intersections of open sets are open: if $U_1, \ldots, U_n \in \tau$, then $U_1 \cap \cdots \cap U_n \in \tau$.