Notation:
- $U \subset R^n$, bounded open set
- $D^k(U) = \{ \omega : U \to \Lambda^k R^n : \text{compactly supported and infinitely differentiable \}}$
- $D_k(U) = D^k(U)'$ is the topological dual space (currents)
- $d : D^k(U) \to D^{k+1}(U)$ is the exterior derivative
The mathematical context is, that $k$-currents $T \in D_k(U)$ provide a generalized notion of $k$-dimensional oriented surface in $\Bbb R^n$, and the flat norm can be used to get a notion of distance between currents.
Let $B\subset\mathbb{R}^n$ be ball and let $\mathcal{D}_0(B)$ be the set of 0-dimensional currents with support in $B$. Recall the flat norm $\mathbb{F}(T)$ of a current $T\in\mathcal{D}_1(B)$ is defined as: $$ \mathbb{F}(T):=\inf\{\mathbb{M}(T-\partial S)+\mathbb{M}(S):S\in\mathcal{D}_{2}(B)\}.$$
Recall also the definition of point-finite collection of sets: A collection $\mathcal{S}$ of subsets of a space $X$ is said to be point-finite if every point in the space $X$ belongs to at most finitely many members of $\mathcal{S}$.
I'm trying to prove that the set
$$
X:=\{b\in{\mathcal{D}}_{0}(B): \mbox{ there exists a rectifiable 1-current }T : \partial T=b\}
$$
has the following property:
For any point-finite open cover $\mathcal{U}$ of $X$, the set $D=\left\{x \in X: \mathcal{U} \text{ is locally finite at } x \right\}$ is a dense set in $X$. I may allow also to prove it for any countable point-finite open cover $\mathcal{U}$ of $X$ or to find a counterexample.
