Cardinality of a closed uncountable set. I am aware that every closed uncountable subset of reals has cardinality of the continuum. 
It's easy to use this result to prove that the same is true in $\mathbb R ^n$, for each $n\in\omega$. 
Question. What are some generalizations of this result? That is, are there general properties that a separable metric space can have, which guarantee every one of its closed uncountable subset has size of the continuum?
 A: This is too long for a comment, but: we could also ask about extending the "closed" side of the question. That is, in a "reasonable" metric space, what sort of sets do we know have the perfect set property? If this is interesting to you, you might be interested in Descriptive Set Theory!
For example, one result in descriptive set theory is that in any complete separable metric space (the underlying topological space of such a space is called Polish, by the way), every analytic set has the perfect set property. This is proved in Kechris' book, in the chapter "Games People Play" if I recall correctly. The analytic sets are a broad class of sets, properly extending the Borel sets.
Meanwhile, it is consistent with ZFC that coanalytic sets need not have this property! And in fact statements of the form "every set of type $\Gamma$ has the perfect set property" are closely related to large cardinals - roughly speaking, if we assume that the combinatorial structure of the set-theoretic universe is sufficiently rich, then every projective set has the perfect set property! (The projective sets are what you get by starting with the Borel sets and closing off under continuous image and complementation - that they extend the Borel sets properly was proved by Souslin, correcting an error of Lebesgue.) There are a pair of themes here:


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*Regularity properties for sets of reals follow naturally from the determinacy of certain kinds of infinite games.

*Determinacy principles follow from, and in a certain sense imply (we have to be careful here - what I really mean is that they imply the existence of an inner model with large cardinals, or more coarsely the consistency of large cardinals with ZFC), large cardinals - which I'm not going to even try to sketch here, but see e.g. http://projecteuclid.org/download/pdfview_1/euclid.ndjfl/1427202981.)
Descriptive set theory beyond the projective is studied, but I know very little about it.
