Semigroups for which the continuum congruences form finite chains (Q789533)

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.