Alexandroff space

Topological space $X$ is called Alexandroff if the intersection of every family of open sets is open.

Of course every finite topological space is Alexandroff, but there are also bigger Alexandroff spaces. For example let $ℝ$ denote the set of real numbers and let $\tau =\{[a,\mathrm{\infty })|a\in ℝ\}\cup \{(b,\mathrm{\infty })|b\in ℝ\}$. Then $\tau$ is a topology on $ℝ$ and $(ℝ,\tau )$ is an Alexandroff space.

If $X$ is an Alexandroff space and $A\subseteq X$, then we may talk about smallest open neighbourhood of $A$. Indeed, let

$$A^o=\bigcap \{U\subseteq X|U\text{ is open and }A\text{ is contained in }U\}.$$

Then $A^o$ is open.