Martin’s Axiom and ω -resolvability of Baire spaces

Fidel Casarrubias-Segura; Fernando Hernández-Hernández; Angel Tamariz-Mascarúa

Commentationes Mathematicae Universitatis Carolinae (2010)

  • Volume: 51, Issue: 3, page 519-540
  • ISSN: 0010-2628

Abstract

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We prove that, assuming MA, every crowded T 0 space X is ω -resolvable if it satisfies one of the following properties: (1) it contains a π -network of cardinality < 𝔠 constituted by infinite sets, (2) χ ( X ) < 𝔠 , (3) X is a T 2 Baire space and c ( X ) ≤ ℵ 0 and (4) X is a T 1 Baire space and has a network 𝒩 with cardinality < 𝔠 and such that the collection of the finite elements in it constitutes a σ -locally finite family. Furthermore, we prove that the existence of a T 1 Baire irresolvable space is equivalent to the existence of a T 1 Baire ω -irresolvable space, and each of these statements is equivalent to the existence of a T 1 almost- ω -irresolvable space. Finally, we prove that the minimum cardinality of a π -network with infinite elements of a space Seq ( u t ) is strictly greater than ℵ 0 .

How to cite

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Casarrubias-Segura, Fidel, Hernández-Hernández, Fernando, and Tamariz-Mascarúa, Angel. "Martin’s Axiom and $\omega $-resolvability of Baire spaces." Commentationes Mathematicae Universitatis Carolinae 51.3 (2010): 519-540. <http://eudml.org/doc/38148>.

@article{Casarrubias2010,
abstract = {We prove that, assuming MA, every crowded $T_0$ space $X$ is $\omega $-resolvable if it satisfies one of the following properties: (1) it contains a $\pi $-network of cardinality $< \mathfrak \{c\}$ constituted by infinite sets, (2) $\chi (X) < \mathfrak \{c\}$, (3) $X$ is a $T_2$ Baire space and $c(X) \le \aleph _0$ and (4) $X$ is a $T_1$ Baire space and has a network $\mathcal \{N\}$ with cardinality $< \mathfrak \{c\}$ and such that the collection of the finite elements in it constitutes a $\sigma $-locally finite family. Furthermore, we prove that the existence of a $T_1$ Baire irresolvable space is equivalent to the existence of a $T_1$ Baire $\omega $-irresolvable space, and each of these statements is equivalent to the existence of a $T_1$ almost-$\omega $-irresolvable space. Finally, we prove that the minimum cardinality of a $\pi $-network with infinite elements of a space $\operatorname\{Seq\}(u_t)$ is strictly greater than $\aleph _0$.},
author = {Casarrubias-Segura, Fidel, Hernández-Hernández, Fernando, Tamariz-Mascarúa, Angel},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Martin’s Axiom; Baire spaces; resolvable spaces; $\omega $-resolvable spaces; almost resolvable spaces; almost-$\omega $-resolvable spaces; infinite $\pi $-network; Martin's axiom; Baire space; resolvable space; -resolvable space; almost resolvable space; almost--resolvable space; infinite -network},
language = {eng},
number = {3},
pages = {519-540},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Martin’s Axiom and $\omega $-resolvability of Baire spaces},
url = {http://eudml.org/doc/38148},
volume = {51},
year = {2010},
}

TY - JOUR
AU - Casarrubias-Segura, Fidel
AU - Hernández-Hernández, Fernando
AU - Tamariz-Mascarúa, Angel
TI - Martin’s Axiom and $\omega $-resolvability of Baire spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2010
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 51
IS - 3
SP - 519
EP - 540
AB - We prove that, assuming MA, every crowded $T_0$ space $X$ is $\omega $-resolvable if it satisfies one of the following properties: (1) it contains a $\pi $-network of cardinality $< \mathfrak {c}$ constituted by infinite sets, (2) $\chi (X) < \mathfrak {c}$, (3) $X$ is a $T_2$ Baire space and $c(X) \le \aleph _0$ and (4) $X$ is a $T_1$ Baire space and has a network $\mathcal {N}$ with cardinality $< \mathfrak {c}$ and such that the collection of the finite elements in it constitutes a $\sigma $-locally finite family. Furthermore, we prove that the existence of a $T_1$ Baire irresolvable space is equivalent to the existence of a $T_1$ Baire $\omega $-irresolvable space, and each of these statements is equivalent to the existence of a $T_1$ almost-$\omega $-irresolvable space. Finally, we prove that the minimum cardinality of a $\pi $-network with infinite elements of a space $\operatorname{Seq}(u_t)$ is strictly greater than $\aleph _0$.
LA - eng
KW - Martin’s Axiom; Baire spaces; resolvable spaces; $\omega $-resolvable spaces; almost resolvable spaces; almost-$\omega $-resolvable spaces; infinite $\pi $-network; Martin's axiom; Baire space; resolvable space; -resolvable space; almost resolvable space; almost--resolvable space; infinite -network
UR - http://eudml.org/doc/38148
ER -

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