Tightness of compact spaces is preserved by the t -equivalence relation

Oleg Okunev

Commentationes Mathematicae Universitatis Carolinae (2002)

  • Volume: 43, Issue: 2, page 335-342
  • ISSN: 0010-2628

Abstract

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We prove that if there is an open mapping from a subspace of C p ( X ) onto C p ( Y ) , then Y is a countable union of images of closed subspaces of finite powers of X under finite-valued upper semicontinuous mappings. This allows, in particular, to prove that if X and Y are t -equivalent compact spaces, then X and Y have the same tightness, and that, assuming 2 𝔱 > 𝔠 , if X and Y are t -equivalent compact spaces and X is sequential, then Y is sequential.

How to cite

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Okunev, Oleg. "Tightness of compact spaces is preserved by the $t$-equivalence relation." Commentationes Mathematicae Universitatis Carolinae 43.2 (2002): 335-342. <http://eudml.org/doc/248999>.

@article{Okunev2002,
abstract = {We prove that if there is an open mapping from a subspace of $C_p(X)$ onto $C_p(Y)$, then $Y$ is a countable union of images of closed subspaces of finite powers of $X$ under finite-valued upper semicontinuous mappings. This allows, in particular, to prove that if $X$ and $Y$ are $t$-equivalent compact spaces, then $X$ and $Y$ have the same tightness, and that, assuming $2^\{\mathfrak \{t\}\}>\mathfrak \{c\}$, if $X$ and $Y$ are $t$-equivalent compact spaces and $X$ is sequential, then $Y$ is sequential.},
author = {Okunev, Oleg},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {function spaces; topology of pointwise convergence; tightness; compact space; tightness; -equivalence; upper semicontinuous finite-valued mapping},
language = {eng},
number = {2},
pages = {335-342},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Tightness of compact spaces is preserved by the $t$-equivalence relation},
url = {http://eudml.org/doc/248999},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Okunev, Oleg
TI - Tightness of compact spaces is preserved by the $t$-equivalence relation
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 2
SP - 335
EP - 342
AB - We prove that if there is an open mapping from a subspace of $C_p(X)$ onto $C_p(Y)$, then $Y$ is a countable union of images of closed subspaces of finite powers of $X$ under finite-valued upper semicontinuous mappings. This allows, in particular, to prove that if $X$ and $Y$ are $t$-equivalent compact spaces, then $X$ and $Y$ have the same tightness, and that, assuming $2^{\mathfrak {t}}>\mathfrak {c}$, if $X$ and $Y$ are $t$-equivalent compact spaces and $X$ is sequential, then $Y$ is sequential.
LA - eng
KW - function spaces; topology of pointwise convergence; tightness; compact space; tightness; -equivalence; upper semicontinuous finite-valued mapping
UR - http://eudml.org/doc/248999
ER -

References

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  1. Arhangel'skii A.V., The spectrum of frequencies of a topological space and the product operation, Trudy Moskov. Mat. Obshch. 40 (1979), 171-206 Russian English translation: Trans. Moscow Math. Soc. (1981), 40 2 169-199. (1981) MR0550259
  2. Arhangel'skii A.V., Problems in C p -theory, 603-615 Open Problems in Topology J. van Mill and G.M. Reed North-Holland (1990). (1990) 
  3. Arhangel'skii A.V., Topological Function Spaces, Kluwer Acad. Publ. Dordrecht (1992). (1992) MR1485266
  4. van Douwen E.K., The Integers and Topology, 111-167 Handbook of Set-Theoretic Topology K. Kunen and J.E. Vaughan North-Holland Amsterdam (1984). (1984) Zbl0561.54004MR0776622
  5. Engelking R., General Topology, PWN (1977), Warszawa. (1977) Zbl0373.54002MR0500780
  6. Gul'ko S.P., Khmyleva T.E., Compactness is not preserved by the relation of t -equivalence, Matematicheskie Zametki 39 6 (1986), 895-903 Russian English translation: Math. Notes 39 5-6 (1986), 484-488. (1986) MR0855937
  7. Malykhin V.I., On tightness and the Suslin number in exp X and in a product of spaces, Dokl. Akad. Nauk SSSR 203 (1972), 1001-1003 Russian English translation: Soviet Math. Dokl. (1972), 13 496-499. (1972) MR0300241
  8. Okunev O., Weak topology of a dual space and a t -equivalence relation, Matematicheskie Zametki 46 1 53-59 (1989), Russian English translation: Math. Notes 46 1-2 534-536 (1989). (1989) MR1019256
  9. Okunev O., A method for constructing examples of M -equivalent spaces, Topology Appl. 36 157-171 (1990), Correction Topology Appl. 49 191-192 (1993). (1993) Zbl0779.54008MR1068167
  10. Ranchin D., Tightness, sequentiality and closed coverings, Dokl. AN SSSR 32 (1977), 1015-1018 Russian English translation: Soviet Math. Dokl. (1977), 18 1 196-199. (1977) Zbl0371.54010MR0436074
  11. Tkachuk V.V., Duality with respect to the functor C p and cardinal invariants of the type of the Souslin number, Matematicheskie Zametki 37 3 (1985), 441-445 Russian English translation: Math. Notes, 37 3 (1985), 247-252. (1985) MR0790433
  12. Tkachuk V.V., Some non-multiplicative properties are l -invariant, Comment. Math. Univ. Carolinae 38 1 (1997), 169-175. (1997) Zbl0886.54005MR1455481

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