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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: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.

The classical computational spectral problem takes $\Omega =\mathcal{B}({\ell }^2(\mathbb{N}))$, reads the matrix entries $⟨A{e}_j,e_i⟩$, and asks for $\mathrm{Sp}(A)$ in the Hausdorff distance or Attouch--Wets topology. Its Solvability complexity index is three for general bounded operators.

A computational problem in the SCI hierarchy is a quadrupleHere $\Omega$ is the primary set of inputs, $\Lambda$ is the set of permitted evaluation functions, $(\mathcal{M},d)$ is the output metric space, and $\Xi :\Omega \to \mathcal{M}$ is the problem function. The Solvability complexity index is defined from the minimum height of a tower of algorithms that computes $\Xi$ from finite subsets of $\Lambda$.

For the classical computational spectral problem, takeandSince the spectrum of a bounded operator is a nonempty compact subset of the complex numbers, one may take $\mathcal{M}$ to be the nonempty compact subsets of $\mathbb{C}$ with the Hausdorff distance. The Attouch--Wets topology gives an equivalent convenient formulation on bounded spectral sets and also extends naturally to unbounded closed sets. This choice of $\mathcal{M}$ makes convergence mean convergence of the whole spectrum as a set, including both the absence of persistent spectral pollution and the approximation of every genuine spectral point.

For a densely defined closed operator $S$, define its lower normThe closed-range theorem givesThereforeand

We next show that $\mu (T(z))$ is accessible through the matrix-entry evaluations. Put $L_n=\mathrm{span}\{e_1,\dots ,e_n\}$ and defineBecause the canonical span is a core for both operators,

For fixed $m,n$ and grid point $z$, formIts entries belong to $\Lambda$. The smallest singular value of $M_{m,n}(z)$ increases as $m\to \infty$ to ${\nu }_n(T(z))$. Applying the same construction to the conjugate-transposed entries gives ${\nu }_n(T(z{)}^{\ast })$. Thus two nested finite-matrix limits determine $\mu (T(z))$.

The assumed gap continuity of $T$ and the identitymake the corresponding lower-norm tests stable under movement of $z$. The rational grids $G_n$ become dense on every bounded disk. Hence an arithmetic algorithm can, on $G_n\cap {\overline{B}}_n(0)$, use the two finite-section levels above and a vanishing rational tolerance to output all grid cells certified byTaking their closures and letting the mesh and tolerance vanish converges in the Attouch--Wets topology toThe strict existential inequality requires two nested limits, with approximants entering from the prescribed side. In the notation of the arithmetic hierarchy this proves

For the spectrum, equality to zero is the countable intersectionUse the preceding two-level pseudospectral procedure with threshold $1/q$, and add an outer limit $q\to \infty$. Outer approximants remove every point with positive lower norm, while gap continuity and density of the grids retain every zero. Truncating to expanding disks and using vanishing mesh again gives Attouch–Wets convergence. The universal outer intersection reverses the one-sided classification and adds one level, proving