Span and plane separating continua (Q2744636)

All considered spaces are compact metric. Let \(X\) be a continuum with \(\pi_1\) and \(\pi_2\) the natural projections of \(X\times X\) onto \(X\) and with the metric \(d\). The span \(\sigma(X)\) is defined as \(\sup\{\varepsilon>0\): there is a continuum \(Z\subset X\times X\) with \(\pi_1(Z)= \pi_2(Z)\) and \(d(x,y) \geq \varepsilon\) for all \((x,y)\in Z\}\). Upper semicontinuous continuum valued functions are used to compute the lower bound for the span of plane separating continua.