While taking my old dog on his afternoon walk, it pop into my brain what you are asking about. It has been nearly forty years since I thought about Moore’s Axiom 1. But that is what you have reference to. The axiom appears on page 1 of his 1932 Foundations of Point Set Topology. Although he developed the axiom some fifteen years before (even before Hausdorff). That axiom actually defines a developable space long before anyone even considered that concept.
There is absolutely no way to get into a discussion of such a broad topic here. I suggest you get a copy of that book and read it.
But here is what you are going for.
If is an open set and there is a closed set then there is an open set such that .