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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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