Attouch-Wets convergence and Kuratowski convergence on compact sets

Paolo Piccione; Rosella Sampalmieri

Commentationes Mathematicae Universitatis Carolinae (1995)

  • Volume: 36, Issue: 3, page 551-562
  • ISSN: 0010-2628

Abstract

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Let X be a locally connected, b -compact metric space and E a closed subset of X . Let 𝔾 be the space of all continuous real-valued functions defined on some closed subsets of E . We prove the equivalence of the τ a w and τ K c topologies on 𝔾 , where τ a w is the so called Attouch-Wets topology, defined in terms of uniform convergence of distance functionals, and τ K c is the topology of Kuratowski convergence on compacta.

How to cite

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Piccione, Paolo, and Sampalmieri, Rosella. "Attouch-Wets convergence and Kuratowski convergence on compact sets." Commentationes Mathematicae Universitatis Carolinae 36.3 (1995): 551-562. <http://eudml.org/doc/247709>.

@article{Piccione1995,
abstract = {Let $X$ be a locally connected, $b$-compact metric space and $E$ a closed subset of $X$. Let $\mathbb \{G\}$ be the space of all continuous real-valued functions defined on some closed subsets of $E$. We prove the equivalence of the $\{\tau _\{_\{a\!w\}\}\}$ and $\{\tau ^c_\{_\{\!K\}\}\}$ topologies on $\mathbb \{G\}$, where $\tau _\{_\{a\!w\}\}$ is the so called Attouch-Wets topology, defined in terms of uniform convergence of distance functionals, and $\{\tau ^c_\{_\{\!K\}\}\}$ is the topology of Kuratowski convergence on compacta.},
author = {Piccione, Paolo, Sampalmieri, Rosella},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {function spaces; Kuratowski convergence; hyperspaces; Kuratowski convergence on compacta; Attouch-Wets convergence},
language = {eng},
number = {3},
pages = {551-562},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Attouch-Wets convergence and Kuratowski convergence on compact sets},
url = {http://eudml.org/doc/247709},
volume = {36},
year = {1995},
}

TY - JOUR
AU - Piccione, Paolo
AU - Sampalmieri, Rosella
TI - Attouch-Wets convergence and Kuratowski convergence on compact sets
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1995
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 36
IS - 3
SP - 551
EP - 562
AB - Let $X$ be a locally connected, $b$-compact metric space and $E$ a closed subset of $X$. Let $\mathbb {G}$ be the space of all continuous real-valued functions defined on some closed subsets of $E$. We prove the equivalence of the ${\tau _{_{a\!w}}}$ and ${\tau ^c_{_{\!K}}}$ topologies on $\mathbb {G}$, where $\tau _{_{a\!w}}$ is the so called Attouch-Wets topology, defined in terms of uniform convergence of distance functionals, and ${\tau ^c_{_{\!K}}}$ is the topology of Kuratowski convergence on compacta.
LA - eng
KW - function spaces; Kuratowski convergence; hyperspaces; Kuratowski convergence on compacta; Attouch-Wets convergence
UR - http://eudml.org/doc/247709
ER -

References

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  10. Kuratowski K., Topology, Academic Press New York (1966). (1966) Zbl0163.17002MR0217751
  11. Mosco U., Convergence of convex sets and solutions of variational inequalities, Advances in Mathematics 3 (1969), 510-585. (1969) MR0298508
  12. Naimpally S.A., Graph topology for function spaces, Trans. Amer. Math. Soc. 123 (1966), 267-272. (1966) Zbl0151.29703MR0192466
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  14. Sampalmieri R., Kuratowski convergence on compact sets, Atti Sem. Mat. Fis. Univ. Modena 39 (1992), 381-390. (1992) Zbl0770.54016MR1200296

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