Definition 1

A topological structure (or, more briefly, a topology) on a set $X$ is a structure given by a set $\mathscr{D}$ of subsets of $X$, having the following properties (called axioms of topological structures):

(O_I) Every union of sets of $\mathscr{D}$ is a set of $\mathscr{D}$.

(O_{II}) Every finite intersection of sets of $\mathscr{D}$ is a set of $\mathscr{D}$.

The sets of $\mathscr{D}$ are called open sets of the topological structure defined by $\mathscr{D}$ on $X$.