Closed subsets of a CAT(0)2–complex are intrinsically CAT(0)

Abstract

Let $\kappa \le 0$, and let $X$ be a complete, locally finite $CAT\left(\kappa \right)$ polyhedral $2$–complex $X$, each face with constant curvature $\kappa$. Let $E$ be a closed, rectifiably connected subset of $X$ with trivial first singular homology. We show that $E$, under the induced path metric, is a complete $CAT\left(\kappa \right)$space.