Let be a complete manifold, be the set of compact/closed subsets of $M$. Let be a complete metric space.
A map is said to be upper-semicontinuous at , if
for any open neighbourhood , there exists a neighbourhood , such that for all .
for any , and any sequence , the limit set .
Viewed as a multivalued function, let be the graph of . Then is u.s.c. if and only if is a closed graph.
And is said to be lower-semicontinuous at , if
for any open set intersecting , there exists neighbourhood such that for all .
or equally, for any , and any sequence , there exists such that .
Let be the set of diffeomorphisms, and be the closure of transverse homoclinic intersections of stable and unstable manifolds of some hyperbolic periodic points of . Then is lower semicontinuous.
Given . Note that it suffices to consider those points . Let and be the continuations of and for . Pick large enough such that . Then for sufficiently close to , and are close to and . In particular is close to .