JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 15, No.l, June 2002 VARIATION OF ORBIT-COINCIDENCE SETS Anjali Srivastava ABSTRACT. David Gauld [3] proved that in many familiar cases the upper semi-finite topology on the set of closed subsets of a space is the largest topology making the coincidence function continuous, when the collection of functions is given the graph topology. Considering G-spaces and taking the coincidence set to consist of points where orbits coincidence, we obtain G-version of many of his results. 1. Introduction An action of a topological group G on a topological space Y is a continuous map 0 from G x Y to Y satisfying 6(e,y) = y and (91, <92, y)) = 6 1 2,?/),where pi,p2 G and e is the identity of G :a topological space together with a given action is called a G- space. Denote 0(g,y) by g • y. For an element y of a G-space F, the set {gr • j; | p 6 G} denoted by G(y) is called the orbit of y. The collection Y/G of orbits together with the topology coinduced by the map 7r :Y ―> y/G taking y to G(y) is called the orbit space of Y. The map 7r is called the orbit map. It is an open map and becomes a closed map as well when G is compact. If Y is Hausdorff and G is compact, then Y/G is Hausdorff. Each G E G determines a homeomorphism Tg :Y - y defined by Tg(y) — g • y^y EY. The action of (7 on F is called proper if the map from (7 x F to K x Y defined by sending (g,y) to (Tg(y 丄y), g E G^y EY is proper i.e., closed with compact fibres. If G acts on Y properly, then also G/Y is Hausdorff [See 5]. If Received by the editors on April 26, 2002 . 2000 Mathematics Subject Classifications: Primary 37Cxx.. Key words and phrases: Coincidence set, Graph topology, Upper semi-finite topology, ( -space.. 1