A note on span under refinable maps (Q1081156)

Let \(\pi_ 1\) and \(\pi_ 2\) be the projection maps onto the first and second factors of \(X\times X\), where (X,\(\rho)\) is a (metric) continuum. The semispan \(\sigma_ 0(X)\) is the least upper bound of the \(\epsilon\geq 0\) for which there exists a continuum \(Z_{\epsilon}\) in \(X\times X\) with \(\pi_ 2(Z_{\epsilon})\subset \pi_ 1(Z_{\epsilon})\) and \(\rho\) (x,y)\(\geq \epsilon\) for each (x,y) in \(Z_{\epsilon}\). The span \(\sigma\) (X) is the number obtained by also requiring that \(\pi_ 2(Z_{\epsilon})=\pi_ 1(Z_{\epsilon})\). A surjective map \(r: X\times Y\) of continua is refinable if for each \(\epsilon >0\) there exists an \(\epsilon\)-map \(f: X\to Y\), surjection, such that \(d(f,r)<\epsilon\). The author shows that for such r, X has (semi) span 0 if and only if Y has (semi) span 0.