If T is a relation, X
the set of first elements and Y a set containing all the second elements,
T(x) = {y ∈ Y |(x,y) ∈ T} and T−1(y) = {x ∈ X|(x,y) ∈ T}. If T(x) ∩ 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.
