Decomposability of ℝ and dense negative subsets of ℝ in constructive reverse mathematics

Abstract We show that over BISH, the decomposability of $\mathbb{R}$ is equivalent to WLPO. Let $D\subseteq \mathbb{R} \setminus \{0\}$ be a dense negative subset of $\mathbb{R}$. It is shown that over BISH, the decomposability of $D$ is equivalent to $\textbf{MP}^{\lor }$.