On the Lindelöf property of spaces of continuous functions over a Tychonoff space and its subspaces
Commentationes Mathematicae Universitatis Carolinae, Tome 50 (2009) no. 4, pp. 629-635
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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, $Y=X\cup\{p\}$, 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 $l(C_p(Y)^\omega)\le l(C_p(X)^\omega)$ and $\operatorname{ext}(C_p(Y)^\omega)\le \operatorname{ext}(C_p(X)^\omega)$.
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, $Y=X\cup\{p\}$, 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 $l(C_p(Y)^\omega)\le l(C_p(X)^\omega)$ and $\operatorname{ext}(C_p(Y)^\omega)\le \operatorname{ext}(C_p(X)^\omega)$.
@article{CMUC_2009_50_4_a11,
author = {Okunev, Oleg},
title = {On the {Lindel\"of} property of spaces of continuous functions over a {Tychonoff} space and its subspaces},
journal = {Commentationes Mathematicae Universitatis Carolinae},
pages = {629--635},
year = {2009},
volume = {50},
number = {4},
mrnumber = {2583139},
zbl = {1212.54052},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMUC_2009_50_4_a11/}
}
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 SP - 629 EP - 635 VL - 50 IS - 4 UR - http://geodesic.mathdoc.fr/item/CMUC_2009_50_4_a11/ LA - en ID - CMUC_2009_50_4_a11 ER -
%0 Journal Article %A Okunev, Oleg %T On the Lindelöf property of spaces of continuous functions over a Tychonoff space and its subspaces %J Commentationes Mathematicae Universitatis Carolinae %D 2009 %P 629-635 %V 50 %N 4 %U http://geodesic.mathdoc.fr/item/CMUC_2009_50_4_a11/ %G en %F CMUC_2009_50_4_a11
Okunev, Oleg. On the Lindelöf property of spaces of continuous functions over a Tychonoff space and its subspaces. Commentationes Mathematicae Universitatis Carolinae, Tome 50 (2009) no. 4, pp. 629-635. http://geodesic.mathdoc.fr/item/CMUC_2009_50_4_a11/