Exercise 21, § 4

Let M be a compact space endowed with a semigroup structure such that (i) the mapping $(x, y) \to xy$ of $M \times M$ into $M$ is continuous; (ii) for each $a \in M$ the relation $ax = ay$ implies $x = y$.

a) Show that if F is a closed subset of M and $x \in M$ is such that $xF \subset F$, then $xF = F$. [Let $y$ be a cluster point of the sequence $(x^n)_{n \geq 1}$ in M; show that $yF = \bigcap_{n \geq 1} x^n F$, and deduce that $yx F = yF$].

b) Deduce from a) that if, in addition, for each $a \in M$ the relation $xa = ya$ implies $x = y$, then M is a compact topological group. (Start by showing that there is an identity element e; to show that $x \to x^{-1}$ is continuous, argue by reductio ad absurdum, considering an ultrafilter on M which converges to e).