Convergence in compacta and linear Lindelöfness

Aleksander V. Arhangel'skii; Raushan Z. Buzyakova

Commentationes Mathematicae Universitatis Carolinae (1998)

  • Volume: 39, Issue: 1, page 159-166
  • ISSN: 0010-2628

Abstract

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Let X be a compact Hausdorff space with a point x such that X ∖ { x } is linearly Lindelöf. Is then X first countable at x ? What if this is true for every x in X ? We consider these and some related questions, and obtain partial answers; in particular, we prove that the answer to the second question is “yes” when X is, in addition, ω -monolithic. We also prove that if X is compact, Hausdorff, and X ∖ { x } is strongly discretely Lindelöf, for every x in X , then X is first countable. An example of linearly Lindelöf hereditarily realcompact non-Lindelöf space is constructed. Some intriguing open problems are formulated.

How to cite

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Arhangel'skii, Aleksander V., and Buzyakova, Raushan Z.. "Convergence in compacta and linear Lindelöfness." Commentationes Mathematicae Universitatis Carolinae 39.1 (1998): 159-166. <http://eudml.org/doc/248237>.

@article{Arhangelskii1998,
abstract = {Let $X$ be a compact Hausdorff space with a point $x$ such that $X\setminus \lbrace x\rbrace $ is linearly Lindelöf. Is then $X$ first countable at $x$? What if this is true for every $x$ in $X$? We consider these and some related questions, and obtain partial answers; in particular, we prove that the answer to the second question is “yes” when $X$ is, in addition, $\omega $-monolithic. We also prove that if $X$ is compact, Hausdorff, and $X\setminus \lbrace x\rbrace $ is strongly discretely Lindelöf, for every $x$ in $X$, then $X$ is first countable. An example of linearly Lindelöf hereditarily realcompact non-Lindelöf space is constructed. Some intriguing open problems are formulated.},
author = {Arhangel'skii, Aleksander V., Buzyakova, Raushan Z.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {point of complete accumulation; linearly Lindelöf space; local compactness; first countability; $\kappa $-accessible diagonal; point of complete accumulation; linearly Lindelöf space; local compactness; first countability; -accessible diagonal},
language = {eng},
number = {1},
pages = {159-166},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Convergence in compacta and linear Lindelöfness},
url = {http://eudml.org/doc/248237},
volume = {39},
year = {1998},
}

TY - JOUR
AU - Arhangel'skii, Aleksander V.
AU - Buzyakova, Raushan Z.
TI - Convergence in compacta and linear Lindelöfness
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1998
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 39
IS - 1
SP - 159
EP - 166
AB - Let $X$ be a compact Hausdorff space with a point $x$ such that $X\setminus \lbrace x\rbrace $ is linearly Lindelöf. Is then $X$ first countable at $x$? What if this is true for every $x$ in $X$? We consider these and some related questions, and obtain partial answers; in particular, we prove that the answer to the second question is “yes” when $X$ is, in addition, $\omega $-monolithic. We also prove that if $X$ is compact, Hausdorff, and $X\setminus \lbrace x\rbrace $ is strongly discretely Lindelöf, for every $x$ in $X$, then $X$ is first countable. An example of linearly Lindelöf hereditarily realcompact non-Lindelöf space is constructed. Some intriguing open problems are formulated.
LA - eng
KW - point of complete accumulation; linearly Lindelöf space; local compactness; first countability; $\kappa $-accessible diagonal; point of complete accumulation; linearly Lindelöf space; local compactness; first countability; -accessible diagonal
UR - http://eudml.org/doc/248237
ER -

References

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  9. Hušek M., Topological spaces without κ -accessible diagonal, Comment. Math. Univ. Carolinae 18 (1977), 777-788. (1977) MR0515009
  10. Hušek M., Convergence versus character in compact spaces, Coll. Math. Soc. J. Bolyai 23 (1980), 647-651. (1980) MR0588812
  11. Juhász I., Cardinal Functions, in M. Hušek and J. van Mill, Ed-rs: Recent Progress in General Topology, North-Holland, Amsterdam, 1993. MR1229134
  12. Mischenko A.S., Finally compact spaces, Soviet Math. Dokl. 145 (1962), 1199-1202. (1962) MR0141070
  13. Rudin M.E., Some Conjectures, in: J. van Mill and G.M. Reed, Ed-ors, Open Problems in Topology (1990), pp.184-193, North-Holland, Amsterdam. MR1078646

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