Exercise 27, § 9

Show that in a compact space $X$ every isodyne [§ 2, Exercise 11 d)] infinite open subspace is solvable. [Using Proposition 1 of no. 2, show that there is a base $\mathfrak{B}$ of the topology of $U$ such that card ($\mathfrak{B}$) = card (U); then use Exercise 11 c) of § 2.] Deduce that every locally compact space with no isolated points is solvable [use Exercise 11 d) of § 2]; consequently such a space is not submaximal.