Disk-like products of continua
By a continuum is meant a nondegenerate, compact, connected metric space. A continuum \(X\) is called disk-like if for each \(\varepsilon>0\), there is an \(\varepsilon\)-map \(f\) of \(X\) onto \([0,1]\times[0,1]\), i.e., for each \(x\in[0,1]\times[0,1]\), the diameter of \(f^{-1}(x)\) is less than \(\varepsilon\). The authors cite a result due to \textit{C. Hagopian} [Proc. Am. Math. Soc. 51, 448--452 (1975; Zbl 0308.54025)] which states that if \(X\), \(Y\) are continua and \(X\times Y\) is disk-like, then \(X\), \(Y\) are atriodic and hereditarily unicoherent. They now establish more facts about such a disk-like product. One says that a space \(X\) is contractible with respect to another space \(Z\) if every map of \(X\) to \(Z\) is inessential. The main results of the paper are: Theorem 7. If \(X\) and \(Y\) are continua such that \(X\times Y\) is disk-like, then \(X\) is contractible with respect to any \(\roman{ANR}\) and each subcontinuum of \(X\) is contractible with respect to \(S^1\). Theorem 9. If \(X\) and \(Y\) are continua such that \(X\times Y\) is disk-like, then \(X\) is a tree-like continuum that is hereditarily weakly confluent.
scientific article; zbMATH DE number 5144522
scientific article