= Attouch--Wets topology
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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:
$$
\sup_{x\in B}|d(x,F_i)-d(x,F)|\longrightarrow0
\qquad(B\text{ bounded}).
$$
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 $F_n=\overline{\operatorname{span}}\{e_k:k\geq n\}$. 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.
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