*Accepted Paper*

**Inserted:** 14 may 2013

**Last Updated:** 18 feb 2014

**Journal:** ESIAM COCV

**Year:** 2013

**Abstract:**

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

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