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Every Filter is Homeomorphic to Its Square

Medini A., Zdomskyy L.

Bulletin of the Polish Academy of Sciences. Mathematics

2016

Vol. 64, no. 1

63--67

We show that every filter F on ω, viewed as a subspace of 2ω, is homeomorphic to F2. This generalizes a theorem of van Engelen, who proved that this holds for Borel filters.

Topological Structure of Non-separable Sigma-locally Compact Convex Sets

Banakh I., Banakh T., Koshino K.

Bulletin of the Polish Academy of Sciences. Mathematics

2013

Vol. 61, no 2

149--153

For an infinite cardinal K let l2 (K) be the linear hull of the standard othonormal base of the Hilbert space l2(K) of density K. We prove that a non-separable convex subset X of density K = dens(X) in a locally convex linear metric space is homeomorphic to the space.[...]

Construction of an Uncountable Difference between Φ(B) and Φƒ(B)

Campbell J., Svanson D.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

Vol. 56, no 2

121-130

We construct a set B and homeomorphism ƒ where ƒ and ƒ[sup]-1 have property N such that the symmetric difference between the sets of density points and of ƒ-density points of B is uncountable.

On homeomorphisms of the density type topologies

Filipczak M., Hejduk J., Wilczyński W.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2005

Vol. 45, [Z] 2

151--159

This paper is dealing of the homeomorphisms of the density type topologies introduced in: M. Filipczak, J. Hejduk, "On topologies associated with the Lebesgue measure", Tatra Mountains Math. Publ. 28 (2004), 187-197