On the Projective Cover of an orbit space

Abstract

In this paper, we obtain the projective cover of the orbit space X/G in terms of the orbit space of the projective space of X, when X is a Tychonoff G-space and G is a finite discrete group. An example shows that flniteness of G is needed.

On projective lift and orbit spaces

“…(b) Let / : X -» Y be a dp-epimorphism. Then for a regular closed set H of Y we have C l / ( C l / - 1 Finally, the uniqueness of Ef follows by recalling that a dp-map is a c-map [5, 6G3]. U…”

“…By inducing an action of a discrete group G on EX, where X is a G-space, Azad and Agrawal [1] showed that if G is finite, then E(X/G) is homeomorphic to the orbit space EX/G. An example to show that the above result may fail if G is infinite, is provided in [1].…”

“…By inducing an action of a discrete group G on EX, where X is a G-space, Azad and Agrawal [1] showed that if G is finite, then E(X/G) is homeomorphic to the orbit space EX/G. An example to show that the above result may fail if G is infinite, is provided in [1]. Here in Section 3, by showing the projective lift Ef of a dp-map / from a locally compact space to a space to be an open map, we study the projective cover of an orbit space X/G, when G is an infinite discrete group.…”

“…Clearly For a G-space X, the orbit {g.x \ g £ G} of x in X is denoted by Gx and for A C X, the set {g.a \ a £ A}, where g £ G, is denoted by g.A. Thus if G is discrete, by defining g.T = {g-F \ F £ T} for an T in EX, one induces an action of G on EX [1]. We denote a member of E(X/G) by TGX and assume that the same converges to Gx in XIG.…”