Exercise 12, § 4

a) Let $X$ be a compact space and $G$ a topological group operating continuously on $X$. Suppose that each orbit $A$ with respect to $G$ has a non-empty interior relative to its closure $\overline{A}$. Show that there is at least one compact orbit (consider a minimal element in the set of compact subsets of $X$ which are stable under $G$).

\* b) Give an example of a locally compact group operating continuously on a compact space, such that none of the orbits is compact (§ 2, Exercise 29). \*