Shape invariance of \(\mathbb{N}\)-compactifications
Let \(N\) denote the natural numbers with the discrete topology. In this paper a space is 0-dimensional if it has a base of clopen sets. A 0- dimensional space \(X\) has a unique \(N\)-compactification, \(\nu (X)\), which is characterized by the property that if \(f : X \to N\) is continuous, then there is a unique extension \(\widetilde f : \nu (X) \to N\). The authors use \(N\)-compactifications of 0-dimensional spaces to obtain a new shape invariant for all topological spaces. They give examples to show that the shape and topological classifications are not the same in the realm of 0-dimensional Tikhonov spaces. The authors do give some classes of spaces where it is true that having the same shape is equivalent to being homeomorphic within the given class of spaces.
scientific article; zbMATH DE number 527423
scientific article