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OCA and towers in 𝒫 ( ℕ ) / f i n

Ilijas Farah (1996)

Commentationes Mathematicae Universitatis Carolinae

We shall show that Open Coloring Axiom has different influence on the algebra 𝒫 ( ℕ ) / f i n than on ℕ ℕ / f i n . The tool used to accomplish this is forcing with a Suslin tree.

On a question of Sierpiński

Theodore Slaman (1999)

Fundamenta Mathematicae

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 automorphisms of Boolean algebras embedded in P (ω)/fin

Magdalena Grzech (1996)

Fundamenta Mathematicae

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

Taras O. Banakh, Ostap Chervak, Lubomyr Zdomskyy (2013)

Commentationes Mathematicae Universitatis Carolinae

We prove that for an unbounded metric space X , the minimal character 𝗆 χ ( X ˇ ) of a point of the Higson corona X ˇ of X is equal to 𝔲 if X has asymptotically isolated balls and to max { 𝔲 , 𝔡 } otherwise. This implies that under 𝔲 < 𝔡 a metric space X of bounded geometry is coarsely equivalent to the Cantor macro-cube 2 < ℕ if and only if dim ( X ˇ ) = 0 and 𝗆 χ ( X ˇ ) = 𝔡 . 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 continuity of measurable group representations and homomorphisms

Yulia Kuznetsova (2012)

Studia Mathematica

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 infinite partitions of lines and space

Paul Erdös, Steve Jackson, R. Mauldin (1997)

Fundamenta Mathematicae

Given a partition P:L → ω of the lines in ℝ n , n ≥ 2, into countably many pieces, we ask if it is possible to find a partition of the points, Q : ℝ n → ω , 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...

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