Sequential completeness of subspaces of products of two cardinals

Roman Frič; Nobuyuki Kemoto

Czechoslovak Mathematical Journal (1999)

  • Volume: 49, Issue: 1, page 119-125
  • ISSN: 0011-4642

Abstract

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Let κ be a cardinal number with the usual order topology. We prove that all subspaces of κ 2 are weakly sequentially complete and, as a corollary, all subspaces of ω 1 2 are sequentially complete. Moreover we show that a subspace of ( ω 1 + 1 ) 2 need not be sequentially complete, but note that X = A × B is sequentially complete whenever A and B are subspaces of κ .

How to cite

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Frič, Roman, and Kemoto, Nobuyuki. "Sequential completeness of subspaces of products of two cardinals." Czechoslovak Mathematical Journal 49.1 (1999): 119-125. <http://eudml.org/doc/30470>.

@article{Frič1999,
abstract = {Let $\kappa $ be a cardinal number with the usual order topology. We prove that all subspaces of $\kappa ^2$ are weakly sequentially complete and, as a corollary, all subspaces of $\omega _1^2$ are sequentially complete. Moreover we show that a subspace of $(\omega _1+1)^2$ need not be sequentially complete, but note that $X=A\times B$ is sequentially complete whenever $A$ and $B$ are subspaces of $\kappa $.},
author = {Frič, Roman, Kemoto, Nobuyuki},
journal = {Czechoslovak Mathematical Journal},
keywords = {sequentially continuous; sequentially complete; product space; sequentially continuous; sequentially complete; product space},
language = {eng},
number = {1},
pages = {119-125},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Sequential completeness of subspaces of products of two cardinals},
url = {http://eudml.org/doc/30470},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Frič, Roman
AU - Kemoto, Nobuyuki
TI - Sequential completeness of subspaces of products of two cardinals
JO - Czechoslovak Mathematical Journal
PY - 1999
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 49
IS - 1
SP - 119
EP - 125
AB - Let $\kappa $ be a cardinal number with the usual order topology. We prove that all subspaces of $\kappa ^2$ are weakly sequentially complete and, as a corollary, all subspaces of $\omega _1^2$ are sequentially complete. Moreover we show that a subspace of $(\omega _1+1)^2$ need not be sequentially complete, but note that $X=A\times B$ is sequentially complete whenever $A$ and $B$ are subspaces of $\kappa $.
LA - eng
KW - sequentially continuous; sequentially complete; product space; sequentially continuous; sequentially complete; product space
UR - http://eudml.org/doc/30470
ER -

References

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  1. Sequential envelope and subspaces of the Čech-Stone compactification, In General Topology and its Relations to Modern Analysis and Algebra III (Proc. Third Prague Topological Sympos., 1971), Academia, Praha, 1971, pp. 123–126. (1971) MR0353260
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  3. Sequentially complete spaces, Czechoslovak Math. J. 29 (1979), 287–297. (1979) MR0529516
  4. Sequentially complete spaces, J. Korean Math. Soc. 9 (1972), 39–43. (1972) Zbl0242.54024MR0303508
  5. On sequentially regular convergence spaces, Czechoslovak Math. J. 17 (1967), 232–247. (1967) MR0215277
  6. Products of spaces of ordinal numbers, Top. Appl. 45 (1992), 245–260. (1992) MR1180812
  7. On sequential envelope, In General Topology and its Relations to Modern Analysis and Algebra I (Proc. First Prague Topological Sympos., 1961 ), Publishing House of the Czecoslovak Academy of Sciences, Praha, 1962, pp. 292–294. (1962) MR0175082

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