OF SPACES AS FUNCTION SPACES

Abstract

0. Introduction. Given a topological space X, we can consider the group G(X) of all autohomeomorphisms of X. Much is known about the relationship between X and G(X) for certain restricted classes of the space X; Whittaker [7] has shown that the existence of an isomorphism between any two sufficiently large subgroups of G(X) and G(Y) implies that X and Y are actually homeomorphic, whenever these are both compact, locally Euclidean manifolds, with or without boundary; Fine and Schweigert [1] give a detailed analysis of G(U); recently, Neumann [4], Mekler [3] and Truss [6] have considered in depth the group G(Q). A proven technique when studying arbitrary spaces is to embed them within other spaces about which more is known; thus the study of compact Hausdorff spaces allows for a greater understanding of Tychonov spaces (i.e. those spaces which occur as subspaces of compact Hausdorff spaces). Similarly, Shimrat [5] has shown that every space X can be embedded in a homogeneous superspace. We shall show that every space X embeds as a retract within the space C(G(X), X) of continuous functions from G(X) into X (with suitably defined topologies), and that this embedding has the additional property that every autohomeomorphism of X extends to an autohomeomorphism of C(G(X), X). Moreover, if X is Tychonov, so is C(G(X), X), and our retraction extends to a retraction of /3C(G(X), X) onto /3X.