Aposyndesis and coherence of continua under refinable maps (Q794321)

A map g between metric spaces is said to be an \(\epsilon\)-map if for each y, \(g^{-1}(y)\) is of diameter less than \(\epsilon\). And g is an \(\epsilon\)-refinement of a map f if it is an \(\epsilon\)-map which is \(\epsilon\) close to f. If this is true for each \(\epsilon>0\), f is said to be refinable. Clearly refinable maps will possess some of the properties of homeomorphisms. The author shows refinable maps preserve aposyndesis, semi-aposyndesis, the commutativity of the set function T, mutual aposyndesis, k-coherence, strong unicoherence and hereditary unicoherence. The preservation of strong aposyndesis is an open question. The author shows that if X and Y are continua and f is refinable with \(f(X)=Y\), then for each \(y\in Y\) and each (Wilson) oscillatory set Y[y], there exists a subcontinuum H of X such that \(f^{-1}(y)\subset H\subset f^{-1}(Y[y])\). Kato has shown that refinable maps preserve irreducibility, but leaves open the question: If X is irreducible from a to b, is Y irreducible from f(a) to f(b)? Using the above result on oscillatory sets, the author gets a partial solution to this question: it is if Y is semi-locally connected at each point of a dense subset.