Topology on a set

A topology on a set $X$ is a collection $\tau$ of subsets of $X$ called the open sets, such that:

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 $U_1 \cap \cdots \cap U_n \in \tau$.

Remarks

• A set $X$ together with a topology $\tau$ on $X$ is called a topological space.