Topological spaces with isomorphic homeomorphism groups



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Let G(F) and G(F') denote the respective groups of all homeomorphisms of the topological spaces F and F1 onto themselves topologized under the point open topology. The groups G(F) and G(F') will be said to be isomorphic if there exists a homeomorphism between them which is also an algebraic isomorphism. The purpose of this thesis is to develop the most general approach to the problem: Under what conditions on F and F' does an isomorphism between G(F) and G(F') induce a homeomorphism between F and F'? We define a class of restrictions on F and F' and prove the following theorem. Let F and F' be topological spaces that satisfy the class of restrictions. Let [empty set] be an isomorphism of G(F') onto G(F). For each x in F1, [empty set](G[lowered x]) is the subgroup Gy of a point y of F. The induced map [empty set]: x—> y of F' into F is a homeomorphism.