Exercise 18, § 5

a) Show that in a complete metric space, every perfect set contains a subset homeomorphic to Cantor’s triadic set (Chapter IV, § 2, no. 5) (use Exercise 11 of Chapter IV, § 8).

b) Deduce that, in a complete metric space with a countable base, every perfect set has the power of the continuum and that every closed set and every open set is either countable or has the power of the continuum [use Exercise 11 b)].

c) Show that in an uncountable complete metric space with a countable base, the set of perfect subsets has the power of the continuum. [Prove the assertion first for the compact space $\{0, 1\}^\mathbf{N}$, then use a) and § 2, Exercise 12 b).]

d) Let $X$ be an uncountable complete metric space with a countable base. Show that there is a subset $Z$ of $X$ such that both $Z$ and $X - Z$ have the power of the continuum and such that neither $Z$ nor $X - Z$ contains a non-empty perfect set. [Use b) and c) and the method described in Set Theory, Chapter III, § 6, Exercise 24 a).] If, furthermore, $X$ has no isolated point, show that neither $Z$ nor $X - Z$ is meagre. (If $A$ is a countable intersection of dense open subsets of $X$, show that $A$ contains non-empty perfect sets by using c) and § 2, Exercise 24.) Deduce that in this case $Z$ is not an almost open set in $X$ (Exercise 6; note that the reasoning above shows that if $U$ is any non-empty open subset of $X$, $U \cap Z$ is not meagre).