On the Lindelöf property of spaces of continuous functions over a Tychonoff space and its subspaces

Oleg Okunev

Commentationes Mathematicae Universitatis Carolinae (2009)

  • Volume: 50, Issue: 4, page 629-635
  • ISSN: 0010-2628

Abstract

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We study relations between the Lindelöf property in the spaces of continuous functions with the topology of pointwise convergence over a Tychonoff space and over its subspaces. We prove, in particular, the following: a) if C p ( X ) is Lindelöf, Y = X ∪ { p } , and the point p has countable character in Y , then C p ( Y ) is Lindelöf; b) if Y is a cozero subspace of a Tychonoff space X , then l ( C p ( Y ) ω ) ≤ l ( C p ( X ) ω ) and ext ( C p ( Y ) ω ) ≤ ext ( C p ( X ) ω ) .

How to cite

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Okunev, Oleg. "On the Lindelöf property of spaces of continuous functions over a Tychonoff space and its subspaces." Commentationes Mathematicae Universitatis Carolinae 50.4 (2009): 629-635. <http://eudml.org/doc/35136>.

@article{Okunev2009,
abstract = {We study relations between the Lindelöf property in the spaces of continuous functions with the topology of pointwise convergence over a Tychonoff space and over its subspaces. We prove, in particular, the following: a) if $C_p(X)$ is Lindelöf, $Y=X\cup \lbrace p\rbrace $, and the point $p$ has countable character in $Y$, then $C_p(Y)$ is Lindelöf; b) if $Y$ is a cozero subspace of a Tychonoff space $X$, then $l(C_p(Y)^\omega )\le l(C_p(X)^\omega )$ and $\operatorname\{ext\}(C_p(Y)^\omega )\le \operatorname\{ext\}(C_p(X)^\omega )$.},
author = {Okunev, Oleg},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {pointwise convergence; Lindelöf property; space; Lindelöf property},
language = {eng},
number = {4},
pages = {629-635},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the Lindelöf property of spaces of continuous functions over a Tychonoff space and its subspaces},
url = {http://eudml.org/doc/35136},
volume = {50},
year = {2009},
}

TY - JOUR
AU - Okunev, Oleg
TI - On the Lindelöf property of spaces of continuous functions over a Tychonoff space and its subspaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2009
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 50
IS - 4
SP - 629
EP - 635
AB - We study relations between the Lindelöf property in the spaces of continuous functions with the topology of pointwise convergence over a Tychonoff space and over its subspaces. We prove, in particular, the following: a) if $C_p(X)$ is Lindelöf, $Y=X\cup \lbrace p\rbrace $, and the point $p$ has countable character in $Y$, then $C_p(Y)$ is Lindelöf; b) if $Y$ is a cozero subspace of a Tychonoff space $X$, then $l(C_p(Y)^\omega )\le l(C_p(X)^\omega )$ and $\operatorname{ext}(C_p(Y)^\omega )\le \operatorname{ext}(C_p(X)^\omega )$.
LA - eng
KW - pointwise convergence; Lindelöf property; space; Lindelöf property
UR - http://eudml.org/doc/35136
ER -

References

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  5. Kubiš W., Okunev O., Szeptycki P.J., 10.1016/j.topol.2005.09.009, Topology Appl. 153 (2006), 2574--2590. (2006) Zbl1102.54028MR2243735DOI10.1016/j.topol.2005.09.009
  6. Okunev O., 10.1016/0166-8641(93)90041-B, Topology Appl. 49 (1993), 149--166. (1993) Zbl0796.54026MR1206222DOI10.1016/0166-8641(93)90041-B
  7. Pol R., A theorem on the weak topology of C ( X ) for compact scattered X , Fund. Math. 106 2 (1980), 135--140. (1980) Zbl0444.54010MR0580591
  8. Rogers A., Jayne E. (Eds.), Analytic Sets, Academic Press, London, 1980. Zbl0589.54047

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