Continuous selections on spaces of continuous functions

Angel Tamariz-Mascarúa

Commentationes Mathematicae Universitatis Carolinae (2006)

  • Volume: 47, Issue: 4, page 641-660
  • ISSN: 0010-2628

Abstract

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For a space Z , we denote by ℱ ( Z ) , 𝒦 ( Z ) and ℱ 2 ( Z ) the hyperspaces of non-empty closed, compact, and subsets of cardinality ≤ 2 of Z , respectively, with their Vietoris topology. For spaces X and E , C p ( X , E ) is the space of continuous functions from X to E with its pointwise convergence topology. We analyze in this article when ℱ ( Z ) , 𝒦 ( Z ) and ℱ 2 ( Z ) have continuous selections for a space Z of the form C p ( X , E ) , where X is zero-dimensional and E is a strongly zero-dimensional metrizable space. We prove that C p ( X , E ) is weakly orderable if and only if X is separable. Moreover, we obtain that the separability of X , the existence of a continuous selection for 𝒦 ( C p ( X , E ) ) , the existence of a continuous selection for ℱ 2 ( C p ( X , E ) ) and the weak orderability of C p ( X , E ) are equivalent when X is ℕ -compact. Also, we decide in which cases C p ( X , 2 ) and β C p ( X , 2 ) are linearly orderable, and when β C p ( X , 2 ) is a dyadic space.

How to cite

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Tamariz-Mascarúa, Angel. "Continuous selections on spaces of continuous functions." Commentationes Mathematicae Universitatis Carolinae 47.4 (2006): 641-660. <http://eudml.org/doc/249889>.

@article{Tamariz2006,
abstract = {For a space $Z$, we denote by $\mathcal \{F\}(Z)$, $\mathcal \{K\}(Z)$ and $\mathcal \{F\}_2(Z)$ the hyperspaces of non-empty closed, compact, and subsets of cardinality $\le 2$ of $Z$, respectively, with their Vietoris topology. For spaces $X$ and $E$, $C_p(X,E)$ is the space of continuous functions from $X$ to $E$ with its pointwise convergence topology. We analyze in this article when $\mathcal \{F\}(Z)$, $\mathcal \{K\}(Z)$ and $\mathcal \{F\}_2(Z)$ have continuous selections for a space $Z$ of the form $C_p(X,E)$, where $X$ is zero-dimensional and $E$ is a strongly zero-dimensional metrizable space. We prove that $C_p(X,E)$ is weakly orderable if and only if $X$ is separable. Moreover, we obtain that the separability of $X$, the existence of a continuous selection for $\mathcal \{K\}(C_p(X,E))$, the existence of a continuous selection for $\mathcal \{F\}_2(C_p(X,E))$ and the weak orderability of $C_p(X,E)$ are equivalent when $X$ is $\mathbb \{N\}$-compact. Also, we decide in which cases $C_p(X,2)$ and $\beta C_p(X,2)$ are linearly orderable, and when $\beta C_p(X,2)$ is a dyadic space.},
author = {Tamariz-Mascarúa, Angel},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {continuous selections; Vietoris topology; linearly orderable space; weakly orderable space; space of continuous functions; dyadic spaces; continuous selections; Vietoris topology; linearly orderable space; weakly orderable space},
language = {eng},
number = {4},
pages = {641-660},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Continuous selections on spaces of continuous functions},
url = {http://eudml.org/doc/249889},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Tamariz-Mascarúa, Angel
TI - Continuous selections on spaces of continuous functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 4
SP - 641
EP - 660
AB - For a space $Z$, we denote by $\mathcal {F}(Z)$, $\mathcal {K}(Z)$ and $\mathcal {F}_2(Z)$ the hyperspaces of non-empty closed, compact, and subsets of cardinality $\le 2$ of $Z$, respectively, with their Vietoris topology. For spaces $X$ and $E$, $C_p(X,E)$ is the space of continuous functions from $X$ to $E$ with its pointwise convergence topology. We analyze in this article when $\mathcal {F}(Z)$, $\mathcal {K}(Z)$ and $\mathcal {F}_2(Z)$ have continuous selections for a space $Z$ of the form $C_p(X,E)$, where $X$ is zero-dimensional and $E$ is a strongly zero-dimensional metrizable space. We prove that $C_p(X,E)$ is weakly orderable if and only if $X$ is separable. Moreover, we obtain that the separability of $X$, the existence of a continuous selection for $\mathcal {K}(C_p(X,E))$, the existence of a continuous selection for $\mathcal {F}_2(C_p(X,E))$ and the weak orderability of $C_p(X,E)$ are equivalent when $X$ is $\mathbb {N}$-compact. Also, we decide in which cases $C_p(X,2)$ and $\beta C_p(X,2)$ are linearly orderable, and when $\beta C_p(X,2)$ is a dyadic space.
LA - eng
KW - continuous selections; Vietoris topology; linearly orderable space; weakly orderable space; space of continuous functions; dyadic spaces; continuous selections; Vietoris topology; linearly orderable space; weakly orderable space
UR - http://eudml.org/doc/249889
ER -

References

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