91Ó°ÊÓ

(e) Let \(\mathrm{G}\) act on \(\mathrm{X}\) and define the orbit of \(\mathrm{x} \in \mathrm{X}\) to be the subset $$ G \cdot x=\\{g \cdot x: g \in G\\} $$ of \(\mathrm{X}\). Prove that two orbits Gix, G.y are either disjoint or equal. Deduce that a Ci-set X decomposes inte a union of disjoint subsets.

(b) Let \(X\) be a G-space with \(G\) finite. Prove that the natural projection \(\pi: \mathrm{X} \rightarrow \mathrm{X} / \mathrm{G}\) is a closed mapping.