I am studying topological spaces, and I have seen that there are $3$ main axioms of separation: $\mathrm{T1}$, Hausdorff and normal.
Now, between Hausdorff and normal there is a case where: given any closed set $F\in T$ and any point $x \in T$, there is open set $O$ containing $F$ and a neighborhood $U$ containing $x$, such that $U \cap O = \varnothing $. This axiom of seperation is not the Hausdorff (stronger than Hausdorff) and not normal (weaker than normal).
My question is: is this indeed a different axiom? Or can it be related via some theorem to one of the existing axioms?
Such a space is called (unimaginatively) regular. It is neither weaker nor stronger than Hausdorff. However, a regular $T_0$ space is automatically Hausdorff. A counterexample: a space with more than one point equipped with the trivial (indiscrete) topology is regular, but it's not Hausdorff.