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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.

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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.

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

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