On \({\mathbb{R}}^{\infty}\)-manifolds and \(Q^{\infty}\)-manifolds. II: Infinite deficiency
This paper is a sequel to Topology Appl. 18, 69-79 (1984; Zbl 0568.57014). In this paper, as generalizations of closed sets contained in collared subsets, we introduce the notion of D-sets \((D^*\)-sets) and establish the unknotting theorem for D-sets in a manifold modelled on \({\mathbb{R}}^{\infty}=dir \lim {\mathbb{R}}^ n\) or \(Q^{\infty}=dir \lim Q^ n\), where Q is the Hilbert cube. This yields equality of D-sets, \(D^*\)- sets and infinite (i.e., \({\mathbb{R}}^{\infty}\)- or \(Q^{\infty}\)-) deficient sets. Our theorem corresponds to a weak version of the unknotting theorem for infinite deficient sets proved by V. T. Liem. However, our proof is elementary and short. And we give an alternative proof of the infinite deficient embedding approximation theorem due to Liem. Using Anderson-McCharen's trick, this approximation theorem strengthens our unknotting theorem in the strong form. Moreover, we show that the union of two \({\mathbb{R}}^{\infty}\)- (or \(Q^{\infty}\)-) manifolds meeting in an \({\mathbb{R}}^{\infty}\)- (or \(Q^{\infty}\)-) manifold is also an \({\mathbb{R}}^{\infty}\)- (or \(Q^{\infty}\)-) manifold, and that for any space X, \(X\times {\mathbb{R}}\) is an \({\mathbb{R}}^{\infty}\)- (or \(Q^{\infty}\)-) manifold if and only if so is \(X\times I\). And we have also an alternative proof of the collaring theorem established by Liem. Recently, the author obtained the negative answer of Problem 6-4 in this paper.
closed sets contained in collared subsets
infinite deficient embedding approximation
scientific article; zbMATH DE number 3921542