### Quasi-compactness and decompositions for arbitrary relations

Stanley Wertheimer
1971 Pacific Journal of Mathematics
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
more » ... paces. 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. 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. The elements of J* are the nonempty subsets A of X having the property that if T"\y) Π A is nonempty then T~ι(y) is contained in A. THEOREM Let T on X to Y be a relation, I an indexing set and A, A t in A* for all iel. Then (a) X -A e J* if A is not X, (b) T~ιT{A) = A, (c) T(X-A)