Definition 5.1.1 (Topological Space).label Let $X$ be a non-empty set. A topology over $X$ is a family $\topo \subset 2^{X}$ such that

1. (O1)

   $\emptyset \in \topo$ and $X \in \topo$.

2. (O2)

   For any $U, V \in \topo$, $U \cap V \in \topo$.

3. (O3)

   For any $\seqi{U}\subset \topo$, $\bigcup_{i \in I}U_{i} \in \topo$.

The elements of $\topo$ are known as open sets, and the pair $(X, \topo)$ is known as a topological space.