Exercise 28, § 9

Now prove that in a compact space X, every infinite isodyne open subspace U (I, p. 92, Exercise 11 d)) is solvable. (Using Proposition 1 of I, p. 60, now prove that there exists a basis $\mathfrak{B}$ of the topology of U such that $\mathrm{Card}(\mathfrak{B}) = \mathrm{Card}(U)$; use subsequently Exercise 11 c) of I, p. 92). Deduce that every locally compact space without isolated point is solvable (use Exercise 11 d) of I, p. 92); consequently, such a space is not submaximal.