Semigroups for which the continuum congruences form finite chains
S(X) denotes the semigroup of all continuous selfmaps of the topological space X. Let \({\mathcal A}\) be a nonempty family of subcontinua of X with the property that if \(A\in {\mathcal A}\), \(f\in S(X)\) and f maps A homeomorphically onto B, then \(B\in {\mathcal A}\). We associate with each such family \({\mathcal A}\), a congruence \(\sigma\) (\({\mathcal A})\) on S(X) by identifying two functions if whenever one of them is injective on any \(A\in {\mathcal A}\), then the two functions agree on A. Any congruence of the form \(\sigma\) (\({\mathcal A})\) is referred to as a continuum congruence and the partially ordered family of all continuum congruences on S(X) is denoted by \(Con_{{\mathcal K}}(S(X))\). Various conditions are obtained which are all equivalent to the condition that \(Con_{{\mathcal K}}(S(X))\) forms a finite chain. In particular, those spaces X for which this is the case are characterized within the Peano continua and although there are infinitely many such spaces there are essentially only five different types of such spaces.
scientific article; zbMATH DE number 3845852