Corollary

Every open subset of a numerical space is locally path-connected.

Every point of $\mathbf{R}$ possesses a basis of neighbourhoods formed by intervals; consequently, $\mathbf{R}$ is locally path-connected. According to Proposition 9, the numerical space $\mathbf{R}^n$ is locally path-connected, for every integer $n\geqslant 1$. Moreover, an open subset of a locally path-connected topological space is again path-connected, whence the corollary.