Property of “correct exit” and one limit theorem for semi-Markov processes

A necessary and sufficient condition on weak convergence of a sequence or probability measures in the space $D_{[0,\infty)}(X)$ is formulated in terms of first exit times. The proof of necessity is based on continuity of first exit times and first exit points with respect to the Stone–Skorokhod metric on the set of functions that “correctly exit” from an open set $\Delta\subset X$. A limit theorem for semi-Markov processes is proved as an application.