Edited reply.

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’sAxiom 1. But that is what you have reference to. The axiom appears on page 1 of his 1932Foundations of Point Set Topology. Although he developed the axiom some fifteen years before (even before Hausdorff). That axiom actually defines adevelopablespace 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 .