A remark on refinable maps and calmness

Abstract

It is shown that if r : X -► Y is a refinable map between compacta and Y is calm, then r is a shape equivalence. As a corollary, if r : X -► Y is a refinable map between compacta and either X or Y is Sn-like (n > 1), then r is a shape equivalence, where Sn denotes the n-sphere.

Spaces of discrete shape and c -refinable maps that induce shape equivalences

“…In [16] we generalize the construction for arbitrary spaces, by giving to $Sh(X, Y)$ the inverse limit topology as inverse limit in Top of $\{Sh(X, Y_{2})\}_{2EA}$ where $Sh(X, Y_{\lambda})$ is assumed to have the discrete topology for any $\lambda\in A$ . Using these spaces, we will show in section 2 a generalization of a theorem of Kato ([11], [12]). We prove that any $c$ -refinable map $f:Xarrow Y$ is a shape equivalence provided the induced morphism $S(f)\in Sh(X, Y)$ is isolated.…”