The Rothberger property on C p ( Ψ ( 𝒜 ) , 2 )

Daniel Bernal-Santos

Commentationes Mathematicae Universitatis Carolinae (2016)

  • Volume: 57, Issue: 1, page 83-88
  • ISSN: 0010-2628

Abstract

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A space X is said to have the Rothberger property (or simply X is Rothberger) if for every sequence 𝒰 n : n ω of open covers of X , there exists U n 𝒰 n for each n ω such that X = n ω U n . For any n ω , necessary and sufficient conditions are obtained for C p ( Ψ ( 𝒜 ) , 2 ) n to have the Rothberger property when 𝒜 is a Mrówka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family 𝒜 for which the space C p ( Ψ ( 𝒜 ) , 2 ) n is Rothberger for all n ω .

How to cite

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Bernal-Santos, Daniel. "The Rothberger property on $C_p(\Psi (\mathcal {A}),2)$." Commentationes Mathematicae Universitatis Carolinae 57.1 (2016): 83-88. <http://eudml.org/doc/276758>.

@article{Bernal2016,
abstract = {A space $X$ is said to have the Rothberger property (or simply $X$ is Rothberger) if for every sequence $\langle \,\mathcal \{U\}_n:n\in \omega \,\rangle $ of open covers of $X$, there exists $U_n\in \mathcal \{U\}_n$ for each $n\in \omega $ such that $X = \bigcup _\{n\in \omega \}U_n$. For any $n\in \omega $, necessary and sufficient conditions are obtained for $C_p(\Psi (\mathcal \{A\}),2)^n$ to have the Rothberger property when $\mathcal \{A\}$ is a Mrówka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family $\mathcal \{A\}$ for which the space $C_p(\Psi (\mathcal \{A\}),2)^n\,$ is Rothberger for all $n\in \omega $.},
author = {Bernal-Santos, Daniel},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {function spaces; $C_p(X,Y)$; Rothberger spaces; $\Psi $-space},
language = {eng},
number = {1},
pages = {83-88},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The Rothberger property on $C_p(\Psi (\mathcal \{A\}),2)$},
url = {http://eudml.org/doc/276758},
volume = {57},
year = {2016},
}

TY - JOUR
AU - Bernal-Santos, Daniel
TI - The Rothberger property on $C_p(\Psi (\mathcal {A}),2)$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2016
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 57
IS - 1
SP - 83
EP - 88
AB - A space $X$ is said to have the Rothberger property (or simply $X$ is Rothberger) if for every sequence $\langle \,\mathcal {U}_n:n\in \omega \,\rangle $ of open covers of $X$, there exists $U_n\in \mathcal {U}_n$ for each $n\in \omega $ such that $X = \bigcup _{n\in \omega }U_n$. For any $n\in \omega $, necessary and sufficient conditions are obtained for $C_p(\Psi (\mathcal {A}),2)^n$ to have the Rothberger property when $\mathcal {A}$ is a Mrówka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family $\mathcal {A}$ for which the space $C_p(\Psi (\mathcal {A}),2)^n\,$ is Rothberger for all $n\in \omega $.
LA - eng
KW - function spaces; $C_p(X,Y)$; Rothberger spaces; $\Psi $-space
UR - http://eudml.org/doc/276758
ER -

References

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  1. Arhangel'skiĭ A.V., Topological Function Spaces, Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers, 1992. MR1144519
  2. Bernal-Santos D., The Rothberger property on C p ( X , 2 ) , Topology Appl. 196 (2015), 106–119. Zbl1331.54015MR3422736
  3. Bernal-Santos D., Tamariz-Macarúa Á., The Menger property on C p ( X , 2 ) , Topology Appl. 183 (2015), 110–126. Zbl1312.54007MR3310340
  4. Hurewicz W., 10.1007/BF01216792, Math. Z. 24 (1926), 401–421. Zbl51.0454.02MR1544773DOI10.1007/BF01216792
  5. Just W., Miller W., Scheepers M., Szeptycki J., 10.1016/S0166-8641(96)00075-2, Topology Appl. 73 (1996), 241–266. Zbl0870.03021MR1419798DOI10.1016/S0166-8641(96)00075-2
  6. Hrušák M., Szeptycki P.J., Tamariz-Mascarúa Á., 10.1016/j.topol.2004.09.009, Topology Appl. 148 (2005), no. (1-3), 239–252. MR2118968DOI10.1016/j.topol.2004.09.009
  7. Rothberger F., Eine Verschärfung der Eigenschaft C, Fund. Math. 30 (1938), 50–55. Zbl64.0622.01

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