On embeddings into C p ( X ) where X is Lindelöf

Masami Sakai

Commentationes Mathematicae Universitatis Carolinae (1992)

  • Volume: 33, Issue: 1, page 165-171
  • ISSN: 0010-2628

Abstract

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A.V. Arkhangel’skii asked that, is it true that every space Y of countable tightness is homeomorphic to a subspace (to a closed subspace) of C p ( X ) where X is Lindelöf? C p ( X ) denotes the space of all continuous real-valued functions on a space X with the topology of pointwise convergence. In this note we show that the two arrows space is a counterexample for the problem by showing that every separable compact linearly ordered topological space is second countable if it is homeomorphic to a subspace of C p ( X ) where X is Lindelöf. Other counterexamples for the problem are also given by making use of the Cantor tree. In addition, we remark that every separable supercompact space is first countable if it is homeomorphic to a subspace of C p ( X ) where X is Lindelöf.

How to cite

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Sakai, Masami. "On embeddings into $C_p(X)$ where $X$ is Lindelöf." Commentationes Mathematicae Universitatis Carolinae 33.1 (1992): 165-171. <http://eudml.org/doc/247387>.

@article{Sakai1992,
abstract = {A.V. Arkhangel’skii asked that, is it true that every space $Y$ of countable tightness is homeomorphic to a subspace (to a closed subspace) of $C_p(X)$ where $X$ is Lindelöf? $C_p(X)$ denotes the space of all continuous real-valued functions on a space $X$ with the topology of pointwise convergence. In this note we show that the two arrows space is a counterexample for the problem by showing that every separable compact linearly ordered topological space is second countable if it is homeomorphic to a subspace of $C_p(X)$ where $X$ is Lindelöf. Other counterexamples for the problem are also given by making use of the Cantor tree. In addition, we remark that every separable supercompact space is first countable if it is homeomorphic to a subspace of $C_p(X)$ where $X$ is Lindelöf.},
author = {Sakai, Masami},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {function space; pointwise convergence; linearly ordered topological space; Lindelöf space; Cantor tree; Lindelöf space; pointwise convergence; linearly ordered topological space; Cantor tree},
language = {eng},
number = {1},
pages = {165-171},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On embeddings into $C_p(X)$ where $X$ is Lindelöf},
url = {http://eudml.org/doc/247387},
volume = {33},
year = {1992},
}

TY - JOUR
AU - Sakai, Masami
TI - On embeddings into $C_p(X)$ where $X$ is Lindelöf
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1992
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 33
IS - 1
SP - 165
EP - 171
AB - A.V. Arkhangel’skii asked that, is it true that every space $Y$ of countable tightness is homeomorphic to a subspace (to a closed subspace) of $C_p(X)$ where $X$ is Lindelöf? $C_p(X)$ denotes the space of all continuous real-valued functions on a space $X$ with the topology of pointwise convergence. In this note we show that the two arrows space is a counterexample for the problem by showing that every separable compact linearly ordered topological space is second countable if it is homeomorphic to a subspace of $C_p(X)$ where $X$ is Lindelöf. Other counterexamples for the problem are also given by making use of the Cantor tree. In addition, we remark that every separable supercompact space is first countable if it is homeomorphic to a subspace of $C_p(X)$ where $X$ is Lindelöf.
LA - eng
KW - function space; pointwise convergence; linearly ordered topological space; Lindelöf space; Cantor tree; Lindelöf space; pointwise convergence; linearly ordered topological space; Cantor tree
UR - http://eudml.org/doc/247387
ER -

References

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  1. Arkhangel'skii A.V., Problems in C p -theory, in: J. van Mill and G.M. Reed, Eds., {Open Problems in Topology}, North-Holland, 1990, 601-615. Zbl0994.54020MR1078667
  2. Engelking R., General Topology, Sigma Series in Pure Math. 6, Helderman Verlag, Berlin, 1989. Zbl0684.54001MR1039321
  3. Lutzer D.J., On generalized ordered spaces, Dissertationes Math. 89 (1971). (1971) Zbl0228.54026MR0324668
  4. Mill J. van, Supercompactness and Wallman spaces, Mathematical Centre Tracts 85 (1977). (1977) MR0464160
  5. Mill J. van, Mills C.F., On the character of supercompact spaces, Top. Proceed. 3 (1978), 227-236. (1978) MR0540493

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