# Countable comeager set in an (uncountable) non perfect polish space

Is it possible to have an uncountable and non perfect Polish space with a countable comeager set? Furthermore, is it possible for this space to have a comeager collection of isolated points?

I've been banging my head against the wall to construct such a space for a while now, but coming up with nothing. Maybe I'm missing something obvious.

Let $$C$$ be the usual Cantor set, obtained from the interval $$[0,1]$$ by repeatedly deleting middle thirds. Let $$M$$ be the set of midpoints of the deleted intervals. Then $$C\cup M$$ is an uncountable, compact subset of $$[0,1]$$, and the countable set $$M$$ of isolated points is open and dense in it, hence comeager.

• D'oh! I thought of this but somehow thought $M$ would be uncountable! Thanks for answering my silly question! – Sucker Oct 31 '20 at 2:58