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Local/global uniform approximation of real-valued continuous functions. (English) Zbl 1240.54062
It is shown that Lindelöf spaces $$X$$ have the following approximation property (AP):
Given an additive lattice subgroup $$G$$ of $$C^{*}(X)$$ containing the constant function $$1$$, which is closed with respect to rational multiples and which separates points and closed sets in $$X$$, the set of $$f\in C(X)$$ which coincide locally with some elements from $$G$$ are uniformly dense in $$C(X)$$.
As a consequence, locally compact paracompact spaces have property (AP). On the other hand, an example of a completely metrizable space by Roy is shown not to have (AP).

##### MSC:
 54C35 Function spaces in general topology 41A30 Approximation by other special function classes 54C40 Algebraic properties of function spaces in general topology 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
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