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Compactifications with finite remainders. (English) Zbl 0674.54015
Let X be a locally compact space, and let n be a positive integer. Define \(B_ n(X)=\{f\in C(X):\) for some compact \(K\subseteq X\), \(| f| X\setminus K]| \leq n\). Then \(B_ n(X)\) determines a compactification \(e_ n(X)\) of X. The author proves that the following statements are equivalent: (1) the remainder of \(e_ n(X)\) consists of precisely n points; (2) X has precisely one n-point compactification; and (3) \(B_ n(X)\) is a subalgebra of \(C^*(X)\), but \(B_ m(X)\) is not, for every \(1<m<n\).
Reviewer: J.van Mill
MSC:
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54D40 Remainders in general topology
54C20 Extension of maps
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