  We discuss the global regularity for 2 dimensional minimal sets that are near
a $\T$ set, that is, whether every global minimal set in $\R^n$ that looks like
a $\T$ set at infinity is a $\T$ set or not. The main point is to use the
topological properties of a minimal set at large scale to control its topology
at smaller scales. This is the idea to prove that all 1-dimensional
Almgren-minimal sets in $\R^n$, and all 2-dimensional Mumford-Shah minimal sets
in $\R^3$ are cones. In this article we discuss two types of 2-dimensional
minimal sets: Almgren-minimal set in $\R^3$ whose blow-in limit is a $\T$ set;
topological minimal sets in $\R^4$ whose blow-in limit is a $\T$ set. For the
first one we eliminate an existing potential counterexample that was proposed
by several people, and show that a real counterexample should have a more
complicated topological structure; for the second we construct a potential
example using a Klein bottle.
