Topology

A topology $$\tau$$ on a set $$X$$ is a subset $$\tau \subseteq \mathscr{P}(X)$$ of the power set of $$X$$, i.e. a collection of subsets of $$X$$, which is closed under unions and closed under finite intersections, i.e. such that


 * (Closure under unions) For any subset $$S \subseteq \tau$$ we have $$\bigcup_{U \in S} U \in \tau$$
 * (Closure under finite intersections) For any finite subset $$S \subseteq \tau$$ we have $$\bigcup_{U \in S} U \in \tau$$.

Note that it follows from these axioms that $$\emptyset \in \tau$$ (taking the empty union) and $$X \in \tau$$ (taking the empty intersection).