Some notes

Let M be a complete manifold, \mathcal{K}_M be the set of compact/closed subsets of $M$. Let X be a complete metric space.

A map \phi: X\to \mathcal{K}_M is said to be upper-semicontinuous at x, if
for any open neighbourhood U\supset \phi(x), there exists a neighbourhood V\ni x, such that \phi(x')\subset U for all x' \in V.
or equally,
for any x_n\to x, and any sequence y_n\in \phi(x_n), the limit set \omega(y_n:n\ge 1)\subset \phi(x).
Viewed as a multivalued function, let G(\phi)=\{(x,y)\subset X\times M: y\in\phi(x)\} be the graph of \phi. Then \phi is u.s.c. if and only if G(\phi) is a closed graph.

And \phi is said to be lower-semicontinuous at x, if
for any open set U intersecting \phi(x), there exists neighbourhood V\ni x such that \phi(x')\cap U\neq\emptyset for all x'\in V.
or equally, for any y\in \phi(x), and any sequence x_n\to x, there exists y_n\in \phi(x_n) such that y\in \omega(y_n:n\ge 1).

Let \mathrm{Diff}^r(M) be the set of C^r diffeomorphisms, and H(f) be the closure of transverse homoclinic intersections of stable and unstable manifolds of some hyperbolic periodic points of f. Then H is lower semicontinuous.

Given f_n\to f. Note that it suffices to consider those points x\in W^s(p,f)\pitchfork W^u(q,f). Let p_n and q_n be the continuations of p and q for f_n. Pick \rho large enough such that x\in W^s_\rho(p,f)\pitchfork W^u_\rho(q,f). Then for f_n sufficiently close to f, W^s_\rho(p_n,f_n) and W^u_\rho(q_n,f_n) are C^1 close to W^s_\rho(p,f) and W^u_\rho(q,f). In particular x_n\in W^s_\rho(p_n,f_n)\pitchfork W^u_\rho(q_n,f_n) is close to x.

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