# On embeddings into ${C}_{p}\left(X\right)$ where $X$ is Lindelöf

Commentationes Mathematicae Universitatis Carolinae (1992)

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

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topSakai, 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

top- 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
- Engelking R., General Topology, Sigma Series in Pure Math. 6, Helderman Verlag, Berlin, 1989. Zbl0684.54001MR1039321
- Lutzer D.J., On generalized ordered spaces, Dissertationes Math. 89 (1971). (1971) Zbl0228.54026MR0324668
- Mill J. van, Supercompactness and Wallman spaces, Mathematical Centre Tracts 85 (1977). (1977) MR0464160
- Mill J. van, Mills C.F., On the character of supercompact spaces, Top. Proceed. 3 (1978), 227-236. (1978) MR0540493

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