EDB — 0W3

[0W3]Topics:perfect set.Prerequisites:[0QP]↺↻,[2F2]↺↻.

Suppose $(X,d)$ is a complete metric space. A closed set without isolated points, i.e. consisting only of accumulation points, is called a perfect set. Show that a non-empty perfect set $E$ contained in $X$ must be uncountably infinite. (Find a simple direct proof, using Baire’s Theorem [0VV]↺↻.)