Definition 2

A topological space is a set endowed with a topological structure.

The elements of a topological space are often called points. When a topology has been defined on a set $X$, this set is said to be the set underlying the topological space $X$.

Axiom (O_I) implies in particular that the union of the empty subset of $\mathscr{D}$, i.e. the empty set belongs to $\mathscr{D}$. Axiom (O_{II}) implies that the intersection of the empty subset of $\mathscr{D}$, i.e. the set $X$, belongs to $\mathscr{D}$.

To show that a set $\mathscr{D}$ of subsets of $X$ satisfies (O_{II}), it is often convenient to prove separately that it satisfies the following two axioms, whose conjunction is equivalent to (O_{II}): (O_{II\ a}) The intersection of two sets of $\mathscr{D}$ belongs to $\mathscr{D}$. (O_{II\ b}) $X$ belongs to $\mathscr{D}$.

Examples of topologies. Given any set $X$, the set of subsets of $X$ consisting of $X$ and $\varnothing$ satisfies axioms $(O_1)$ and $(O_{II})$ and therefore defines a topology on $X$. So does the set $\mathfrak{P}(X)$ of all subsets of $X$: the topology it defines is the discrete topology on $X$, and the set $X$ with this topology is called a discrete space.

A covering $(U_i)_{i \in I}$ of a subset $A$ of a topological space $X$ is said to be open if all the $U_i$ are open in $X$.