  In this paper, we study the critical case of the Allard regularity theorem.
Combining with Reifenberg's topological disk theorem, we get a critical
Allard-Reifenberg type regularity theorem. As a main result, we get the
topological finiteness for a class of properly immersed surfaces in
$\mathbb{R}^n$ with finite Willmore energy. Especially, we prove a removability
of singularity of multiplicity one surface with finite Willmore energy and a
uniqueness theorem of the catenoid under no a priori topological finiteness
assumption.
