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Let $X$ be a locally connected, $b$-compact metric space and $E$ a closed subset of $X$. Let $𝔾$ be the space of all continuous real-valued functions defined on some closed subsets of $E$. We prove the equivalence of the ${\tau }_{{}_{aw}}$ and ${\tau }_{{}_K}^c$ topologies on $𝔾$, where ${\tau }_{{}_{aw}}$ is the so called topology, defined in terms of uniform convergence of distance functionals, and ${\tau }_{{}_K}^c$ is the topology of Kuratowski convergence on compacta.

Let $\left(M,g\right)$ be a complete Riemannian manifold, $\mathrm{\Omega }\subset M$ an open subset whose closure is diffeomorphic to an annulus. If $\partial \mathrm{\Omega }$ is smooth and it satisfies a strong concavity assumption, then it is possible to prove that there are at least two geometrically distinct geodesics in $\overline{\mathrm{\Omega }}=\mathrm{\Omega }\bigcup \partial \mathrm{\Omega }$ starting orthogonally to one connected component of $\partial \mathrm{\Omega }$ and arriving orthogonally onto the other one. The results given in [5] allow to obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating...