Let $(X,T)$ be a topological space such that $\forall A\subset X \ \ \exists F_i \in T $ such that $A=\bigcap\limits_iF_i $, is $(X.T) $ Hausdorff

Question

Let $(X,T)$ be a topological space such that $\forall A\subset X \ \exists F_i \in T $ such that $A=\bigcap\limits_iF_i $ how can I show if $(X,T)$ is or not neccesary Hausdorff

Answer 1

If $X$ is an infinite set with the cofinite topology, then any $A\subset X$ can be written as the intersection of open sets containing it, but $X$ is not Hausdorff.

However, any topological space $X$ satisfying your property must be $T_1$: if $x\in X$, then $X\setminus \{x\}$ is an intersection of open sets, hence open.