Given a topological space (X, %i), let H(X, ql) be the class of all homeomorphisms of (X, ca) onto itself. Everett and Ulam [1], [5] posed the following problem. What topologies 'U exist on X such that H(X, at) =H(X, 'U), i.e., these two topological spaces have exactly the same class of mappings as homeomorphisms. Many topologies VU on X such that H(X, 9a) =H(X, V) have been constructed in [3]. However, all topologies constructed in [3] are coarser than the original topology. In general, we cannot reverse the procedure to reconstruct the topologies 9t out of 'U and to prove that they possess the same class of mappings as homeomorphisms, because the topologies 'U constructed in [3] are hard to characterize. Therefore it is natural to ask the question, given a topological space (X, Ca), can we construct topologies 'U'cit such that H(X, 9a) =H(X, V). This paper is devoted to investigating this problem. By X\A, Cl(A) and Int(A), unless otherwise stated, we always mean the complement, closure and the interior of A relative to the original topology cit. We denote the neighborhood system of a point p with respect to 9a by 91,. A family of subsets X in (X, 9a) is called an a-family if the following three conditions are satisfied: (1) The empty set 0 is in X. (2) If NEX then Int(N) =0. (3) If {N1, . .. , N.4 CX and {fi, * * *, fk4 CH(X, Al), then U {fi(Ni): i=1, = * * , k} IC for every k. An s-family X is said to be a strong s-family if Int(Cl(N)) =0 for all N in X. We shall use the symbol Cl (A, V) to denote the closure of A with respect to a new topology V. The following lemma may be derived straightforwardly from the definition of an a-family: