The convention topology definitionis: let $X$ be a set and let $\mathcal{T}$ be a family of subsets of $X$. Then $\mathcal{T}$ is called a topology on $X$ if:
- Both the $\emptyset$ and $X$ are elements of $\mathcal{T}$.
- Any union of elements of $\mathcal{T}$ is an element of $\mathcal{T}$.
- Any intersection of finitely many elements of $\mathcal{T}$ is an element of $\mathcal{T}$.
My question is, if adding one extra condition:
- Every element $i$ of $\mathcal{T}$ also has it's complement $X \setminus i$ in $\mathcal{T}$.
Is there a term for topologies with this condition ?