Subsets of the reals when the Continuum Hypothesis is assumed false

If one assumes that the continuum hypothesis is false then there are subsets of the reals of intermediate cardinality, uncountable but smaller than the continuum. What can be said about the necessary properties of such a set e.g. topological properties other than that it cannot be closed?

Solution 1:

Without additional hypotheses, not too much actually.

It cannot be a Borel set, since uncountable Borel sets have a copy of the Cantor set and thus they have cardinality continuum.

It cannot be of positive measure, since if $A$ is of positive measure, $A-A=\{a-b\mid a,b\in A\}$ contains an interval, and thus $A$ had to be of size continuum. Similarly if it isn't meager, then it cannot have the Baire property.

It cannot be a continuous image of a Borel set (or analytic), since those also have copies of the Cantor set.

It might be the complement of an analytic set, but if our set is of cardinality $>\aleph_1$, then it won't be, because uncountable co-analytic sets must have size $\aleph_1$ or $2^{\aleph_0}$.

It might be measurable, since it is consistent that every set of size less than $2^{\aleph_0}$ has measure zero. Similarly it might be meager.

There are many many properties, but the above list should be a reasonable start. Perhaps reading about Martin's Axiom, which have consequences that "most" regularity properties hold for sets of size $<2^{\aleph_0}$ by making them "small" (null sets, meager sets, etc.) and learn what sort of things might happen or might not happen.