On the $\sigma$-countable compactness of spaces of continuous functions with the set-open topology
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 3, pp. 251-260
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For a Tychonoff space $X$, we obtain a criterion of the $\sigma$-countable compactness of the space of continuous real-valued functions $C(X)$ with the set-open topology. In particular, for extremally disconnected space $X$, we prove that the space $C_\lambda(X)$ is a $\sigma$-countably compact space if and only if $X$ is a pseudocompact space, the set $X(P)$ of all $P$-points of $X$ is dense in $X$, and the family $\lambda$ consists of finite subsets of $X(P)$.
Keywords:
set-open topology, $\sigma$-countably compact space, extremally disconnected space, $P$-point, space of continuous functions.
@article{TIMM_2013_19_3_a25,
author = {A. V. Osipov and E. G. Pytkeev},
title = {On the $\sigma$-countable compactness of spaces of continuous functions with the set-open topology},
journal = {Trudy Instituta matematiki i mehaniki},
pages = {251--260},
publisher = {mathdoc},
volume = {19},
number = {3},
year = {2013},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TIMM_2013_19_3_a25/}
}
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A. V. Osipov; E. G. Pytkeev. On the $\sigma$-countable compactness of spaces of continuous functions with the set-open topology. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 3, pp. 251-260. http://geodesic.mathdoc.fr/item/TIMM_2013_19_3_a25/