O prvním Hilbertově problému (Hypotéza kontinua a axióm výběru)
O riešení niektorých nerozhodnutelných topologických problémov
OCA and towers in
We shall show that Open Coloring Axiom has different influence on the algebra than on . The tool used to accomplish this is forcing with a Suslin tree.
On a problem of Nogura about the product of Fréchet-Urysohn -spaces
Assuming Martin’s Axiom, we provide an example of two Fréchet-Urysohn -spaces, whose product is a non-Fréchet-Urysohn -space. This gives a consistent negative answer to a question raised by T. Nogura.
On a proof of the Erdős-Monk theorem.
On a question of H. Kraljevic.
On a question of Sierpiński
There is a set U of reals such that for every analytic set A there is a continuous function f which maps U bijectively to A.
On a theorem of Baumgartner and Weese
On a weak Kelley-Morse theory of classes
On automorphisms of Boolean algebras embedded in P (ω)/fin
We prove that, under CH, for each Boolean algebra A of cardinality at most the continuum there is an embedding of A into P(ω)/fin such that each automorphism of A can be extended to an automorphism of P(ω)/fin. We also describe a model of ZFC + MA(σ-linked) in which the continuum is arbitrarily large and the above assertion holds true.
On character of points in the Higson corona of a metric space
We prove that for an unbounded metric space , the minimal character of a point of the Higson corona of is equal to if has asymptotically isolated balls and to otherwise. This implies that under a metric space of bounded geometry is coarsely equivalent to the Cantor macro-cube if and only if and . This contrasts with a result of Protasov saying that under CH the coronas of any two asymptotically zero-dimensional unbounded metric separable spaces are homeomorphic.
On Ciesielski's problems.
On compact spaces carrying Radon measures of uncountable Maharam type
If Martin’s Axiom is true and the continuum hypothesis is false, and X is a compact Radon measure space with a non-separable space, then there is a continuous surjection from X onto .
On continuity of measurable group representations and homomorphisms
Let G be a locally compact group, and let U be its unitary representation on a Hilbert space H. Endow the space ℒ(H) of bounded linear operators on H with the weak operator topology. We prove that if U is a measurable map from G to ℒ(H) then it is continuous. This result was known before for separable H. We also prove that the following statement is consistent with ZFC: every measurable homomorphism from a locally compact group into any topological group is continuous.
On decomposing the plane into connected or one-to-one curves
On Fuchs' problem 76.
On independence of the relations of epimorphy and embeddability on the variety of all lattices.
On infinite partitions of lines and space
Given a partition P:L → ω of the lines in , n ≥ 2, into countably many pieces, we ask if it is possible to find a partition of the points, , so that each line meets at most m points of its color. Assuming Martin’s Axiom, we show this is the case for m ≥ 3. We reduce the problem for m = 2 to a purely finitary geometry problem. Although we have established a very similar, but somewhat simpler, version of the geometry conjecture, we leave the general problem open. We consider also various generalizations...
On intersections of sets of positive Lebesgue measure
On Luzin spaces