The purpose of this note is to describe a functor which provides a framework for certain constructions in topology. It is related to the sets \( (E,\pi,B,X,q) \) described in [1960] and is particularly adapted to discussing the limit of repeated modifications of triangulable spaces. Roughly speaking, one forms a space \( X\Delta K \) by replacing each top dimensional simplex of a complex \( K \) with a copy of a space \( X \). If in addition there are mappings on the spaces \( X \), \( K \), these induce a mapping on the new space \( X\Delta K \).
It has been called to my attention that several authors have considered analogous functors (though not as far as I know, in written form). This is not surprising inasmuch as \( X \Delta K \) is defined just as the Whitney sum of two bundles.
Though the principal applications of this functor are to be found elsewhere, in a paper by Frank Raymond and the author [1960, 1963] and a forthcoming paper by the author, three famous examples [Pontrjagin 1930; Boltyanskii 1951; Kolmogoroff 1937] are given as applications in the last section. Two of these are in dimension theory proper, but the third is essentially about transformation groups.
It is hoped that the reader will find our description of Boltyanskii’s example easier than the original, as a simpler, more homogeneous version is given. In addition, in our version of Kolmogoroff’s example, the group acts without fixed points. This answers a question raised by Anderson [1957].