On the Wallman-Frink compactification of 0-dimensional spaces and shape

Abstract

Here SF denotes the category whose objects are the pairs (X,P)where P is a metrizable ANR-space and X is a closed subset of P, and the morphisms between two objects (X,P) and (Y,Q) are the homotopy classes of mutations f:U(X,P)→V(Y,Q) (where U(X,P) and V(Y,Q) are the complete open neighborhood systems of X in P and Y in Q respectively). So two objects of SF are isomorphic if and only if they have the same shape in the sense of Fox (or Marde sic). The author constructs a covariant functor T from SF to the category C0 of all compact 0-dimensional spaces and continuous maps. This functor allows him to obtain new shape invariants in the class of metrizable spaces. Using this functor T he also constructs new contravariant functors to the the category of metrizable spaces and continuous maps and to the category of groups and homomorphisms. In order to construct T he uses the space of quasicomponents QX of a metrizable space X . Actually he uses the 0-dimensional compactification β0(QX) of QX. The space β0(QX) can be viewed as the 0-dimensional analogue of the Stone-Cech compactification. As a theorem he proves that two 0-dimensional metrizable spaces are of the same shape if and only if they are homeomorphic. This is a generalization in the metric case of a similar result for paracompacta due to G. Kozlowski and the reviewer [Fund. Math. 83 (1974), no. 2, 151-154] because there are metrizable spaces X such that ind(X)=0 but the covering dimension dim(X)>0 .