  In this note we show the characteristic function of every indecomposable set
$F$ in the plane is $BV$ equivalent to the characteristic function a closed set
$\mathbb{F}$, i.e.
$||\mathbb{1}_{F}-\mathbb{1}_{\mathbb{F}}||_{BV(\mathbb{R}^2)}=0$. We show by
example this is false in dimension three and above. As a corollary to this
result we show that for every $\epsilon>0$ a set of finite perimeter $S$ can be
approximated by a closed subset $\mathbb{S}_{\epsilon}$ with finitely many
indecomposable components and with the property that $H^1(\partial^M
\mathbb{S}_{\epsilon}\backslash \partial^M S)=0$ and
$||\mathbb{1}_{S}-\mathbb{1}_{\mathbb{S}_{\epsilon}}||_{BV(\mathbb{R}^2)}<\epsilon$.
We apply this corollary to give a short proof that locally quasiminimizing sets
in the plane are $BV_l$ extension domains.
