Topological space/Definition

A topological space ${\displaystyle {}(X,{\mathcal {T}})}$ consists of a set ${\displaystyle {}X}$ together with a subset ${\displaystyle {}{\mathcal {T}}}$ of the power set of ${\displaystyle {}X}$ that satisfies the following structural conditions (the subsets ${\displaystyle {}U\subseteq X}$ belonging to ${\displaystyle {}{\mathcal {T}}}$ are called open).

1. The empty set and the set ${\displaystyle {}X}$ are open.
2. The intersection of finitely many open sets is again open; that is, if ${\displaystyle {}U_{1},\ldots ,U_{n}\in {\mathcal {T}}}$, then also ${\displaystyle {}U_{1}\cap \ldots \cap U_{n}\in {\mathcal {T}}}$.
3. The union of arbitrary many open sets is again open; that is, if ${\displaystyle {}U_{i}\in {\mathcal {T}}}$ for every ${\displaystyle {}i\in I}$ (of an arbitrary index set ${\displaystyle {}I}$), then also ${\displaystyle {}\bigcup _{i\in I}U_{i}\in {\mathcal {T}}}$.