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Extraresolvability and cardinal arithmetic

Ofelia Teresa Alas, Salvador García-Ferreira, Artur Hideyuki Tomita (1999)

Commentationes Mathematicae Universitatis Carolinae

Following Malykhin, we say that a space X is extraresolvable if X contains a family 𝒟 of dense subsets such that | 𝒟 | > Δ ( X ) and the intersection of every two elements of 𝒟 is nowhere dense, where Δ ( X ) = min { | U | : U is a nonempty open subset of X } is the dispersion character of X . We show that, for every cardinal κ , there is a compact extraresolvable space of size and dispersion character 2 κ . In connection with some cardinal inequalities, we prove the equivalence of the following statements: 1) 2 κ < 2 κ + , 2) ( κ + ) κ is extraresolvable and...

Homogeneity and rigidity in Erdös spaces

Klaas P. Hart, Jan van Mill (2018)

Commentationes Mathematicae Universitatis Carolinae

The classical Erdös spaces are obtained as the subspaces of real separable Hilbert space consisting of the points with all coordinates rational or all coordinates irrational, respectively. One can create variations by specifying in which set each coordinate is allowed to vary. We investigate the homogeneity of the resulting subspaces. Our two main results are: in case all coordinates are allowed to vary in the same set the subspace need not be homogeneous, and by specifying different sets for different...

Irreducibility of inverse limits on intervals

David Ryden (2000)

Fundamenta Mathematicae

A procedure for obtaining points of irreducibility for an inverse limit on intervals is developed. In connection with this, the following are included. A semiatriodic continuum is defined to be a continuum that contains no triod with interior. Characterizations of semiatriodic and unicoherent continua are given, as well as necessary and sufficient conditions for a subcontinuum of a semiatriodic and unicoherent continuum M to lie within the interior of a proper subcontinuum of M.

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