Continua of constant distances in span theory (Q1076363)

Let \(f: X\to Y\) be a mapping between metric spaces. The author defines the span \(\sigma\) (f), the semispan \(\sigma_ 0(f)\), and, if X is connected, the surjective span \(\sigma^*(f)\) and the surjective semispan \(\sigma^*_ 0(f)\) of the mapping f as the least upper bound of the set of reals \(\alpha\) with the following property: there exist nonempty connected sets \(C_{\alpha}\subset X\times X\) such that dist[f(x),f(y)]\(\geq \alpha\) for \((x,y)\in C_{\alpha}\) and (\(\sigma)\) \(p_ 1(C)=p_ 2(C)\); \((\sigma_ 0)\) \(p_ 1(C)\supset p_ 2(C)\); resp. \((\sigma^*)\) \(p_ 1(C)=p_ 2(C)=X\); \((\sigma^*_ 0)\) \(p_ 1(C)=X\), where \(p_ 1\) and \(p_ 2\) denote the standard projections of \(X\times X\) onto X. The concept of the span of a mapping was introduced by \textit{W. T. Ingram} [ibid. 77, 99-107 (1972; Zbl 0244.54023)] as a generalization of the notion of the span of a space due to the present author [Fundam. Math. 55, 199-214 (1964; Zbl 0142.398)], who also introduced other kinds of spans [Pac. J. Math. 64, 207-215 (1976; Zbl 0309.54028)]. The main result is the following. If \(f: X\to Y\) is a mapping between compact metric spaces, \(\tau =\sigma\), \(\sigma_ 0\) or, if X is connected, \(\tau =\sigma\), \(\sigma_ 0\), \(\sigma^*\), \(\sigma^*_ 0\), and \(0\leq \alpha \leq \tau (f)\), then there exists a nonempty continuum \(C_{\alpha}\subset X\times X\) with \(dist[f(x),f(y)]=\alpha\) for \((x,y)\in C_{\alpha}\) and condition (\(\tau)\) is satisfied. The paper also contains some corollaries and applications of the result.