Tychonoff expansions in which a given set is open

Every topology is Tychonoff and crowded (i.e., without isolated points). For a space 〈X,T〉, let I(A) denote the set of isolated points of the subspace X⧹A and define J(A)=I(A)∩A. We prove that: (1) There exists a T′⊇T such that A∈T′ if and only if A admits an expansion family (Definition 3.4); (2) There exists T′⊇T such that A∈T′ and T′|A=T|A if and only if J(A)=∅; and (3) Every T′⊇T satisfies intT′A=intTA if and only if A=intTA∪J(A) and A admits no nonempty partial expansion families.