I have been thinking about the following question: for which topological spaces $X$ are all perfect subspaces of $X$ uncountable, where perfect means closed with no isolated points. As long as $X$ is $T_1$, we know that perfect sets are at least infinite.
One sufficient condition is for a space to be completely Baire, meaning that every closed subspace is Baire. This works since countable sets are meager inside perfect sets, but in a completely Baire space closed sets are nonmeager in themselves. By the Baire Category Theorem this covers complete metric spaces like $\mathbb{R}^n$ (or any Polish space) and locally compact Hausdorff spaces. In these last classes of topological spaces it can actually proved that perfect sets have size continuum.
The counterexample to keep in mind is $\mathbb{Q}$ as a subspace of the reals. This is a metric space which is perfect but countable.
So, I was wondering if anybody knew more about this question, or had some interesting examples. I would be especially curious to know of weaker conditions that show perfect sets have size $\mathfrak{c}$.
Best Answer
In the realm of metrisable spaces, a space $X$ is completely Baire (a.k.a. hereditarily Baire) iff it contains no closed homeomorphic copy of $\Bbb Q$, i.e. a closed perfect set that is countable (the only perfect countable spaces are copies of $\Bbb Q$ within metric spaces) This is due to Hurewicz. So for that class the problem is solved already. (The combined properties of being completely Baire and metrisable do not imply in general that the space is completely metrisable, but it seems close to it).
But you ask for general spaces, in that the class of Čech-complete (aka topologically complete) spaces, i.e. all spaces that can be embedded as a $G_\delta$ subspace into a compact Hausdorff space, or which are Tychonoff and a $G_\delta$ in $\beta X$. This is a class of spaces that includes locally compact Hausdorff spaces and completely metrisable spaces and which are hereditarily Baire. So there we have that all perfect subsets of $X$ are uncountable, and it's a natural and well-studied class of spaces.
But I doubt if such a condition is in general necessary, there are probaby hereditarily Baire spaces that are not topologically complete and maybe we have to look to related classes defined by "topological games" like the Choquet game and the Banach-Mazur game to find a characterisation (if a nice one exists at all) of spaces without a countable perfect subspace.