  In this article we treat two closely related problems: 1) the upper semi
continuity property for Almgren minimal sets in regions with regular boundary,
which guanrantees that the uniqueness property is well defined; and 2) the
Almgren (resp. topological) uniqueness property for all the 2-dimensional
Almgren (resp. topological) minimal cones in $\mathbb{R}^3$. As proved in
\cite{2T}, when several 2-dimensional Almgren (resp. topological) minimal cones
are measure and Almgren (resp. topological) sliding stable, and Almgren (resp.
topological) unique, the almost orthogonal union of them stays minimal. As
consequence, the results of this article, together with the measure and sliding
stability properties proved in \tb{\cite{stablePYT} and \cite{stableYXY}},
permit us to use all known 2-dimensional minimal cones in $\mathbb{R}^n$ to
generate new families of minimal cones by taking their almost orthogonal
unions. The upper semi continuity property is also helpful in various
circumstances: when we have to carry on arguments using Hausdorff limits and
some properties do not pass to the limit, the upper semi continuity can serve
as a link. As an example, it plays a very important role throughout \cite{2T}.
