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If (X, T, tt) is a transformation group with locally compact phase group T, there is a standard way to induce a transformation group on C(X, Y) endowed with the compact-open topology, where y is a uniform space. In this paper, we consider the case where y is a topological group G. The reverse construction under certain conditions is also considered. If (X, T, tt) is a transformation group with locally compact phase group T, there is a standard way [3, 1.68] to induce a transformation group on C(X, Y) endowed with the compact-open topology, where Y is a uniform space. England and Lanier [1] have applied this to the case where Y is the space R of all real numbers and have shown that under certain conditions this construction can be reversed and that many dynamical statements about (X, T, tt) can be faithfully reflected in C(X). The purpose of this paper is to show that many results obtained in [1] can also be obtained under a more general setting. For a topological space X and a topological group G, let C(X, G) be the topological group of all continuous functions from X into G endowed with the compact-open topology and with the pointwise multiplication. The identity element of the group C(X, G) is denoted by /0 which maps every x in X into e, the identity element of G. For each p in X, let CV(X, G) be composed of functions/such that/(/>)=e. The groups C(X, G) and CB(X, G) are discussed in [6], It is pointed out there that G is isomorphic to C(X, G)ICP(X, G) for each p. In our investigation we adopt some notations which are similar to those used in [1] and [2] whenever possible. Throughout this note all transformation groups are supposed to have a locally compact phase group, and the letter G will stand for an arbitrarily chosen but fixed topological group. All topological spaces considered here are Hausdorff. For notations and definitions not given, the reader is referred to [3] and [5]. Presented to the Society, January 26, 1973 under the title A note on the functional transformation group (C(X, G), T); received by the editors June 15, 1972. AMS (MOS) subject classifications (1970). Primary 54H20, 54H15; Secondary 22A05.

Erratum to ‘‘On isomorphic groups and homeomorphic spaces”