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Keywords:
AP space; WAP space; pseudoradial space; radial space; product; compact space; submaximal space; function space
AP space; WAP space; pseudoradial space; radial space; product; compact space; submaximal space; function space
Summary:
Several remarks on the properties of approximation by points (AP) and weak approximation by points (WAP) are presented. We look in particular at their behavior in product and at their relationships with radiality, pseudoradiality and related concepts. For instance, relevant facts are: \noindent (a) There is in ZFC a product of a countable WAP space with a convergent sequence which fails to be WAP. \noindent (b) $C_p$ over $\sigma$-compact space is AP. Therefore AP does not imply even pseudoradiality in function spaces, while it implies Fréchet-Urysohn property in compact spaces. \noindent (c) WAP and AP do not coincide in $C_p$.
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