EQUIVALENCE OF INTRINSIC SHAPE, BASED ON \(\mathcal{V}\)-CONTINUOUS FUNCTIONS, AND SHAPE

Abstract

In this paper is given a direct proof that the intrinsic shape category \(InSh\) constructed with continuous functions over coverings, is equivalent to original shape category \(Sh\) of Borsuk obtained by embedding compact metric spaces in Hilbert cube \(Q\). The functor \(Sh \to InSh\) is established taking a fundamental sequence \((\bar{f}_n)\) from \(X\) to \(Y\) in the sense of Borsuk, and by associating to the continuous function \(\bar{f}_n : Q \to Q\) mapping some neighborhood of \(X\) into a union of the members of a covering \(\mathcal{V}\) of \(Y\), a \(\mathcal{V}\)-continuous function \(f_n : X \to Y\), and forming the proximate sequence \((f_n)\) in the sense of N. Shekutkovski, Top. Proc. 39 (2012).