5
votes
2answers
296 views

If every nonseparable metric space contains a sequence of subsets with no convergent subsequence, does the Continuum Hypothesis hold?

If every nonseparable metric space contains a sequence of subsets with no convergent subsequence, does the Continuum Hypothesis hold? The answer is negative, and in the interests of self-contained ...
3
votes
1answer
97 views

Ultrafilters of weight $\aleph_2$ in Sacks model

It is well-known that in Sacks model there are P-points and even Ramsey ultrafilters, but what the usual (i.e. findable in the literature) proofs for these facts do is proving that ground model ...
16
votes
2answers
464 views

Is the notion of fixed point property for topological spaces an absolute notion?

Recall that a topological space $X$ has the fixed point property (FPP) if any continuous function $f: X\to X$ has a fixed point. Is the notion of FPP for topological spaces an absolute notion? More ...
5
votes
1answer
201 views

Forcing over the poset of nonempty open subsets of a nice topological space

Is there anything sensible to be said concerning a notion of forcing given by the poset of nonempty open subsets of the sort of topological space that comes up in ($e.g.$ algebraic) topology? If so, ...
8
votes
2answers
675 views

If Q is a subset of the plane of size less than continuum, then does every closed F in Q extend to a closed connected G in the plane with the same trace on Q? (Or is this independent of ZFC?)

This question arises in connection with this MO question and especially with Sergei Ivanov's wonderful answer, which showed that for any countable set $Q\subset\mathbb{R}^2$ and every closed set ...