Special sets of reals and weak forms of normality on Isbell--Mrówka spaces

Vinicius de Oliveira Rodrigues; Victor dos Santos Ronchim; Paul J. Szeptycki

Commentationes Mathematicae Universitatis Carolinae (2023)

  • Volume: 64, Issue: 1, page 109-126
  • ISSN: 0010-2628

Abstract

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We recall some classical results relating normality and some natural weakenings of normality in Ψ -spaces over almost disjoint families of branches in the Cantor tree to special sets of reals like Q -sets, λ -sets and σ -sets. We introduce a new class of special sets of reals which corresponds to the corresponding almost disjoint family of branches being ℵ 0 -separated. This new class fits between λ -sets and perfectly meager sets. We also discuss conditions for an almost disjoint family 𝒜 being potentially almost-normal (pseudonormal), in the sense that 𝒜 is almost-normal (pseudonormal) in some c.c.c. forcing extension.

How to cite

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de Oliveira Rodrigues, Vinicius, dos Santos Ronchim, Victor, and Szeptycki, Paul J.. "Special sets of reals and weak forms of normality on Isbell--Mrówka spaces." Commentationes Mathematicae Universitatis Carolinae 64.1 (2023): 109-126. <http://eudml.org/doc/299482>.

@article{deOliveiraRodrigues2023,
abstract = {We recall some classical results relating normality and some natural weakenings of normality in $\Psi $-spaces over almost disjoint families of branches in the Cantor tree to special sets of reals like $Q$-sets, $\lambda $-sets and $\sigma $-sets. We introduce a new class of special sets of reals which corresponds to the corresponding almost disjoint family of branches being $\aleph _0$-separated. This new class fits between $\lambda $-sets and perfectly meager sets. We also discuss conditions for an almost disjoint family $\mathcal \{A\}$ being potentially almost-normal (pseudonormal), in the sense that $\mathcal \{A\}$ is almost-normal (pseudonormal) in some c.c.c. forcing extension.},
author = {de Oliveira Rodrigues, Vinicius, dos Santos Ronchim, Victor, Szeptycki, Paul J.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Isbell–Mrówka spaces; almost disjoint families; almost-normal; weak $\lambda $-set},
language = {eng},
number = {1},
pages = {109-126},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Special sets of reals and weak forms of normality on Isbell--Mrówka spaces},
url = {http://eudml.org/doc/299482},
volume = {64},
year = {2023},
}

TY - JOUR
AU - de Oliveira Rodrigues, Vinicius
AU - dos Santos Ronchim, Victor
AU - Szeptycki, Paul J.
TI - Special sets of reals and weak forms of normality on Isbell--Mrówka spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2023
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 64
IS - 1
SP - 109
EP - 126
AB - We recall some classical results relating normality and some natural weakenings of normality in $\Psi $-spaces over almost disjoint families of branches in the Cantor tree to special sets of reals like $Q$-sets, $\lambda $-sets and $\sigma $-sets. We introduce a new class of special sets of reals which corresponds to the corresponding almost disjoint family of branches being $\aleph _0$-separated. This new class fits between $\lambda $-sets and perfectly meager sets. We also discuss conditions for an almost disjoint family $\mathcal {A}$ being potentially almost-normal (pseudonormal), in the sense that $\mathcal {A}$ is almost-normal (pseudonormal) in some c.c.c. forcing extension.
LA - eng
KW - Isbell–Mrówka spaces; almost disjoint families; almost-normal; weak $\lambda $-set
UR - http://eudml.org/doc/299482
ER -

References

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