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A note on condensations of C p ( X ) onto compacta

Aleksander V. Arhangel'skii; Oleg I. Pavlov

Commentationes Mathematicae Universitatis Carolinae (2002)

  • Volume: 43, Issue: 3, page 485-492
  • ISSN: 0010-2628

Abstract

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A condensation is a one-to-one continuous mapping onto. It is shown that the space C p ( X ) of real-valued continuous functions on X in the topology of pointwise convergence very often cannot be condensed onto a compact Hausdorff space. In particular, this is so for any non-metrizable Eberlein compactum X (Theorem 19). However, there exists a non-metrizable compactum X such that C p ( X ) condenses onto a metrizable compactum (Theorem 10). Several curious open problems are formulated.

How to cite

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Arhangel'skii, Aleksander V., and Pavlov, Oleg I.. "A note on condensations of $C_p(X)$ onto compacta." Commentationes Mathematicae Universitatis Carolinae 43.3 (2002): 485-492. <http://eudml.org/doc/248983>.

@article{Arhangelskii2002,
abstract = {A condensation is a one-to-one continuous mapping onto. It is shown that the space $C_p(X)$ of real-valued continuous functions on $X$ in the topology of pointwise convergence very often cannot be condensed onto a compact Hausdorff space. In particular, this is so for any non-metrizable Eberlein compactum $X$ (Theorem 19). However, there exists a non-metrizable compactum $X$ such that $C_p(X)$ condenses onto a metrizable compactum (Theorem 10). Several curious open problems are formulated.},
author = {Arhangel'skii, Aleksander V., Pavlov, Oleg I.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {condensation; compactum; network; Lindelöf space; topology of pointwise convergence; $\sigma $-compact space; Eberlein compactum; Corson compactum; Borel set; monolithic space; tightness; condensation; compact space; topology of pointwise convergence; cardinal invariants},
language = {eng},
number = {3},
pages = {485-492},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A note on condensations of $C_p(X)$ onto compacta},
url = {http://eudml.org/doc/248983},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Arhangel'skii, Aleksander V.
AU - Pavlov, Oleg I.
TI - A note on condensations of $C_p(X)$ onto compacta
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 3
SP - 485
EP - 492
AB - A condensation is a one-to-one continuous mapping onto. It is shown that the space $C_p(X)$ of real-valued continuous functions on $X$ in the topology of pointwise convergence very often cannot be condensed onto a compact Hausdorff space. In particular, this is so for any non-metrizable Eberlein compactum $X$ (Theorem 19). However, there exists a non-metrizable compactum $X$ such that $C_p(X)$ condenses onto a metrizable compactum (Theorem 10). Several curious open problems are formulated.
LA - eng
KW - condensation; compactum; network; Lindelöf space; topology of pointwise convergence; $\sigma $-compact space; Eberlein compactum; Corson compactum; Borel set; monolithic space; tightness; condensation; compact space; topology of pointwise convergence; cardinal invariants
UR - http://eudml.org/doc/248983
ER -

References

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  13. Tkachenko M.G., Bicompacta that are continuous images of sets everywhere dense in the product of spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. 27:10 (1979), 797-802. (1979) MR0603151
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