A fixed point theorem for Whitney blocks
A continuum is a nonempty compact connected metric space. For a continuum \(X\), let \(C(X)\) denote the hyperspace of all continua in \(X\) with the Hausdorff metric. A map is a continuous function. A Whitney map for \(C(X)\) is a map \(\mu: C(X)\to [0,1]\) such that (a) \(\mu(\{x\})= 0\) for every \(x\in X\), (b) if \(A,B\in C(X)\) and \(A\subsetneq B\), then \(\mu(A)< \mu(B)\), and \(\mu(X)= 1\). A Whitney block on \(C(X)\) (respectively, a Whitney level in \(C(X)\)) is a set of the form \(\mu^{-1}([s,t])\) (respectively, \(\mu^{-1}(t)\)), where \(0\leq s\leq t\leq 1\). It is known that Whitney blocks in \(C(X)\) are continua. For a continuum \(X\), \(X\) has a zero surjective semispan if whenever \(Z\) is a continuum in \(X\times X\) such that \(\pi_1(Z)= X\), then \(Z\) intersects the diagonal in \(X\times X\), where \(\pi_1\) denotes the first projection of \(X\times X\) onto \(X\). A continuum \(X\) is said to have the fixed point property provided that every map \(f\) from \(X\) into \(X\) has a fixed point, i.e., a point \(x\in X\) such that \(f(x)= x\). The authors prove the following theorem and related corollaries concerning the fixed point property: Theorem 3.1 Let \(X\) be a continuum with zero surjective semispan. If \(\mu\) is a Whitney map for \(C(X)\) and \(0\leq s\leq t\leq 1\), then \(\mu^{-1}([s,t])\) has the fixed point property. Note that Whitney blocks and Whitney levels in \(C(X)\) are continua. Since it is known that a Whitney map exists for \(C(X)\), it follows: Corollary 3.2. If \(X\) is a continuum with zero surjective semispan, then \(C(X)\) has the fixed point property. Since it is known that each arc-like continuum has zero surjective semispan, it follows: Corollary 3.3. For an arc-like continuum \(X\), each Whitney block in \(C(X)\) has the fixed point property. A map \(f: Y\to X\) is universal provided \(f\) has a coincidence with every map from \(Y\) into \(X\); this means that for any map \(g: Y\to X\) there exists a point \(p\in Y\) such that \(f(p)= g(p)\). If \(f: Y\to X\) is a universal map, then \(f\) is surjective and \(X\) has the fixed point property. Concerning universal maps, the authors prove the following theorem: Theorem 4.1. Let \(X\) be a continuum with zero surjective semispan. If \(f: Y\to X\) is a map from a continuum \(Y\) onto \(X\), then the induced map \(\widehat f: C(Y)\to C(X)\) is universal, where the induced map \(\widehat f: C(Y)\to C(X)\) means the map defined by \(\widehat f(A)= f(A)\) for every \(A\in C(Y)\).
scientific article; zbMATH DE number 1879127
scientific article