We study variational problems of the form $$\inf \{\lambda_k(\Omega):
\Omega\ \textup{open in}\ \R^m,\ T(\Omega) \le 1 \},$$ where
$\lambda_k(\Omega)$ is the $k$'th eigenvalue of the Dirichlet
Laplacian acting in $L^2(\Omega)$, and where $T$ is a non-negative set
function defined on the open sets in $\R^m$, which is invariant under
isometries, additive on disjoint families of open sets, and is such
that the ball with $T(B)=1$ is a minimiser for $k=1$. Upper bounds are
obtained for the number of components of any bounded minimiser if $T$
satisfies a scaling relation. For example we show that if $T$ is
Lebesgue measure and if $k\le m+1$ then any bounded minimiser has at
most $7$ components.