Metrization of function spaces with the Fell topology

Hanbiao Yang

Commentationes Mathematicae Universitatis Carolinae (2012)

  • Volume: 53, Issue: 2, page 307-318
  • ISSN: 0010-2628

Abstract

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For a Tychonoff space X , let ↓ C F ( X ) be the family of hypographs of all continuous maps from X to [ 0 , 1 ] endowed with the Fell topology. It is proved that X has a dense separable metrizable locally compact open subset if ↓ C F ( X ) is metrizable. Moreover, for a first-countable space X , ↓ C F ( X ) is metrizable if and only if X itself is a locally compact separable metrizable space. There exists a Tychonoff space X such that ↓ C F ( X ) is metrizable but X is not first-countable.

How to cite

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Yang, Hanbiao. "Metrization of function spaces with the Fell topology." Commentationes Mathematicae Universitatis Carolinae 53.2 (2012): 307-318. <http://eudml.org/doc/246992>.

@article{Yang2012,
abstract = {For a Tychonoff space $X$, let $\downarrow \{\rm C\}_F(X)$ be the family of hypographs of all continuous maps from $X$ to $[0,1]$ endowed with the Fell topology. It is proved that $X$ has a dense separable metrizable locally compact open subset if $\downarrow \{\rm C\}_F(X)$ is metrizable. Moreover, for a first-countable space $X$, $\downarrow \{\rm C\}_F(X)$ is metrizable if and only if $X$ itself is a locally compact separable metrizable space. There exists a Tychonoff space $X$ such that $\downarrow \{\rm C\}_F(X)$ is metrizable but $X$ is not first-countable.},
author = {Yang, Hanbiao},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {space of continuous maps; Fell topology; hyperspace; metrizable; hypograph; separable; first-countable; function space; hyperspace; hypograph; Fell topology; metrizable; first countable},
language = {eng},
number = {2},
pages = {307-318},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Metrization of function spaces with the Fell topology},
url = {http://eudml.org/doc/246992},
volume = {53},
year = {2012},
}

TY - JOUR
AU - Yang, Hanbiao
TI - Metrization of function spaces with the Fell topology
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2012
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 53
IS - 2
SP - 307
EP - 318
AB - For a Tychonoff space $X$, let $\downarrow {\rm C}_F(X)$ be the family of hypographs of all continuous maps from $X$ to $[0,1]$ endowed with the Fell topology. It is proved that $X$ has a dense separable metrizable locally compact open subset if $\downarrow {\rm C}_F(X)$ is metrizable. Moreover, for a first-countable space $X$, $\downarrow {\rm C}_F(X)$ is metrizable if and only if $X$ itself is a locally compact separable metrizable space. There exists a Tychonoff space $X$ such that $\downarrow {\rm C}_F(X)$ is metrizable but $X$ is not first-countable.
LA - eng
KW - space of continuous maps; Fell topology; hyperspace; metrizable; hypograph; separable; first-countable; function space; hyperspace; hypograph; Fell topology; metrizable; first countable
UR - http://eudml.org/doc/246992
ER -

References

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  8. Yang Z., Zhang B., 10.1007/s10114-012-0030-6, Acta Math. Sinica, English Ser. 28 (2012), 57–66. MR2863750DOI10.1007/s10114-012-0030-6
  9. Yang Z., Zhou X., 10.1016/j.topol.2006.12.013, Topology Appl. 154 (2007), 1737–1747. Zbl1119.54010MR2317076DOI10.1016/j.topol.2006.12.013
  10. Zhang Y., Yang Z., Hyperspaces of the regions below of upper semi-continuous maps on non-compact metric spaces, Advances in Math. in China 39 (2010), 352–360 (Chinese). MR2724454
  11. McCoy R.A., Ntanyu I., Properties C ( X ) with the epi-topology, Bollettion U.M.I. (7)6-B(1992), 507–532. MR1191951

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