Quasi-compactness and decompositions for arbitrary relations
Stanley Wertheimer · Pacific Journal of Mathematics · 1971
If T is a relation, X the set of first elements and Y a set containing all the second elements, T(x) = {yeY\(x f y)e T] and T~\y) = {x e X | (a?, y) e T}.If T(x) n T(y) is nonempty implies that T(x) ~ T(y), the relation T is semi-single-valued (ssv).Every ssv surjection defines a decomposition of X into point inverses and a decomposition of Y into point images.G. T. Whyburn has analyzed the ssv surjection T on X to Y in terms of these decomposition spaces and the natural mappings onto these spaces.He discusses quasi-compactness for ssv relations.It is the purpose of this paper to extend Whyburn's analysis to include all relations.2* Decompositions* Let P(X) denote the power set of X.DEFINITION 2.1.Let T on X to Y be a relation.Define AT on P(X) to P(Y) by ΔT(A) = T{A) Γ) T(X -A), A* the collection of nonempty subsets of X for which AT(A) is empty, and A the collection of all minimal members of J* with respect to the partial ordering defined by set inclusion.