Topological Space

A topological space is a H-Set $X$ equipped with a subset $\mathcal{O}({X})\subseteq\mathcal{P}(X)$ of the Power Set of $X$ such that:

$\emptyset\in\mathcal{O}({X})$
$X\in\mathcal{O}({X})$
for all $U,V\in\mathcal{O}({X})$ , $(U\cap V)\in\mathcal{O}({X})$
for all $I$ -indexed families $U_{i}$ with $U_{i}\in\mathcal{O}({X})$ , $\bigcup_{i:I}U_{i}\in\mathcal{O}({X})$

Questions