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 $x$.