A topological space is a set $X$ and a collection $\Omega$ of subsets of $X$ such that:
- $\emptyset \in \Omega$ and $X \in \Omega$
- The union of any collection of $\Omega$ is in $\Omega$
- The intersection of any finite collection of sets in $\Omega$ is again in $\Omega$
My question is, is the set $\Omega$ essentially the powerset of $X$? Or is the powerset of $X$ just a special case of a suitable collection of subsets of $X$?