Attouch--Wets topology

For nonempty closed subsets $F_i,F$ of a metric space, Attouch--Wets topology convergence means uniform convergence of their distance functions on every bounded set:It captures approximation of closed sets across each bounded observation region, even when the sets themselves are unbounded.

If $F_n$ decrease to a nonempty intersection $F$ in a proper metric space, then $F_n\to F$ in this sense. For a fixed $x$, almost-minimizing points in $F_n$ stay in one closed bounded ball, using any fixed point of $F$ as a comparison. Compactness of that ball and nestedness show $d(x,F_n)\uparrow d(x,F)$. These distance functions are continuous, so Dini's theorem makes convergence uniform on each compact ball.

This fails in an infinite-dimensional Hilbert space. In ${\ell }^2$, put $F_n=\overline{\mathrm{span}}\{e_k:k\ge n\}$. Their intersection is $\{0\}$, but $d(e_n,F_n)=0$ and $d(e_n,\{0\})=1$. The error on the unit ball stays at least1. Pointwise distance convergence of nested closed sets does not supply uniform convergence on bounded sets without an appropriate compactness hypothesis.