On powers of Lindelöf spaces

Isaac Gorelic

Commentationes Mathematicae Universitatis Carolinae (1994)

  • Volume: 35, Issue: 2, page 383-401
  • ISSN: 0010-2628

Abstract

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We present a forcing construction of a Hausdorff zero-dimensional Lindelöf space X whose square X 2 is again Lindelöf but its cube X 3 has a closed discrete subspace of size 𝔠 + , hence the Lindelöf degree L ( X 3 ) = 𝔠 + . In our model the Continuum Hypothesis holds true. After that we give a description of a forcing notion to get a space X such that L ( X n ) = ℵ 0 for all positive integers n , but L ( X ℵ 0 ) = 𝔠 + = ℵ 2 .

How to cite

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Gorelic, Isaac. "On powers of Lindelöf spaces." Commentationes Mathematicae Universitatis Carolinae 35.2 (1994): 383-401. <http://eudml.org/doc/247569>.

@article{Gorelic1994,
abstract = {We present a forcing construction of a Hausdorff zero-dimensional Lindelöf space $X$ whose square $X^2$ is again Lindelöf but its cube $X^3$ has a closed discrete subspace of size $\{\mathfrak \{c\}\}^+$, hence the Lindelöf degree $L(X^3) = \{\mathfrak \{c\}\}^+ $. In our model the Continuum Hypothesis holds true. After that we give a description of a forcing notion to get a space $X$ such that $L(X^n) = \aleph _0$ for all positive integers $n$, but $L(X^\{\aleph _0\}) = \{\mathfrak \{c\}\}^+ = \aleph _2$.},
author = {Gorelic, Isaac},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {forcing; topology; products; Lindelöf; Lindelöf space},
language = {eng},
number = {2},
pages = {383-401},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On powers of Lindelöf spaces},
url = {http://eudml.org/doc/247569},
volume = {35},
year = {1994},
}

TY - JOUR
AU - Gorelic, Isaac
TI - On powers of Lindelöf spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1994
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 35
IS - 2
SP - 383
EP - 401
AB - We present a forcing construction of a Hausdorff zero-dimensional Lindelöf space $X$ whose square $X^2$ is again Lindelöf but its cube $X^3$ has a closed discrete subspace of size ${\mathfrak {c}}^+$, hence the Lindelöf degree $L(X^3) = {\mathfrak {c}}^+ $. In our model the Continuum Hypothesis holds true. After that we give a description of a forcing notion to get a space $X$ such that $L(X^n) = \aleph _0$ for all positive integers $n$, but $L(X^{\aleph _0}) = {\mathfrak {c}}^+ = \aleph _2$.
LA - eng
KW - forcing; topology; products; Lindelöf; Lindelöf space
UR - http://eudml.org/doc/247569
ER -

References

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  1. Shelah S., On some problems in general topology, preprint, 1978. Zbl0847.54004MR1367138
  2. Juhász I., Cardinal Functions II, in: K. Kunen and J.E. Vaughan, eds., Handbook of Set-theoretic Topology, North-Holland, Amsterdam, 1984. MR0776621
  3. Hajnal A., Juhàsz I., Lindelöf spaces à la Shelah, Coll. Mat. Soc. Bolyai, Budapest, 1978. 
  4. Gorelic I., The Baire Category and forcing large Lindelöf spaces with points G δ , Proceedings Amer. Math. Soc. 118 (1993), 603-607. (1993) MR1132417
  5. Juhász I., Cardinal Functions, in: M. Hušek and J. van Mill, eds., Recent Progress in General Topology, North-Holland, 1992. MR1229134
  6. Przymusinski T.C., Normality and paracompactness in finite and countable cartesian products, Fund. Math. 105 (1980), 87-104. (1980) Zbl0438.54021MR0561584
  7. Kunen K., Set Theory, North-Holland, Amsterdam, 1980. Zbl0960.03033MR0597342

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