Definition 1

A proper partial mapping from X into Y is a correspondence $f= (\Gamma ,X,Y)$ (E, II, p. 10, Def. 2) between X and Y such that

(i) The graph Γ is functional;

(ii) The domain of definition of $f$ is an open subset U of X;

(iii) The mapping $x\mapsto f(x)$ of U into Y is proper.

The identity mapping of X is a proper partial mapping of X into X. Let Z be a locally compact topological space and $f$ (resp. $g$) a proper partial mapping of X into Y (resp. of Y into Z). Then the composite correspondence $g\circ f$ (E, II, p. 11, Def. 6) is a proper partial mapping of X into Z (TG, I, p. 72, Prop. 3, and p. 73, Prop. 5).