## Some notes

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 .

or equally,

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 .

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