Local/global uniform approximation of real-valued continuous functions

Hager, Anthony W.. "Local/global uniform approximation of real-valued continuous functions." Commentationes Mathematicae Universitatis Carolinae 52.2 (2011): 283-291. <http://eudml.org/doc/246920>.

@article{Hager2011,
abstract = {For a Tychonoff space $X$, $C(X)$ is the lattice-ordered group ($l$-group) of real-valued continuous functions on $X$, and $C^\{*\}(X)$ is the sub-$l$-group of bounded functions. A property that $X$ might have is (AP) whenever $G$ is a divisible sub-$l$-group of $C^\{*\}(X)$, containing the constant function 1, and separating points from closed sets in $X$, then any function in $C(X)$ can be approximated uniformly over $X$ by functions which are locally in $G$. The vector lattice version of the Stone-Weierstrass Theorem is more-or-less equivalent to: Every compact space has AP. It is shown here that the class of spaces with AP contains all Lindelöf spaces and is closed under formation of topological sums. Thus, any locally compact paracompact space has AP. A paracompact space failing AP is Roy’s completely metrizable space $\Delta $.},
author = {Hager, Anthony W.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {real-valued function; Stone-Weierstrass; uniform approximation; Lindelöf space; locally in; spaces; uniform approximation; extension of continuous functions; lattice groups in },
language = {eng},
number = {2},
pages = {283-291},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Local/global uniform approximation of real-valued continuous functions},
url = {http://eudml.org/doc/246920},
volume = {52},
year = {2011},
}

TY - JOUR
AU - Hager, Anthony W.
TI - Local/global uniform approximation of real-valued continuous functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2011
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 52
IS - 2
SP - 283
EP - 291
AB - For a Tychonoff space $X$, $C(X)$ is the lattice-ordered group ($l$-group) of real-valued continuous functions on $X$, and $C^{*}(X)$ is the sub-$l$-group of bounded functions. A property that $X$ might have is (AP) whenever $G$ is a divisible sub-$l$-group of $C^{*}(X)$, containing the constant function 1, and separating points from closed sets in $X$, then any function in $C(X)$ can be approximated uniformly over $X$ by functions which are locally in $G$. The vector lattice version of the Stone-Weierstrass Theorem is more-or-less equivalent to: Every compact space has AP. It is shown here that the class of spaces with AP contains all Lindelöf spaces and is closed under formation of topological sums. Thus, any locally compact paracompact space has AP. A paracompact space failing AP is Roy’s completely metrizable space $\Delta $.
LA - eng
KW - real-valued function; Stone-Weierstrass; uniform approximation; Lindelöf space; locally in; spaces; uniform approximation; extension of continuous functions; lattice groups in
UR - http://eudml.org/doc/246920
ER -