#### Volume 21, issue 4 (2021)

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Closed subsets of a $\mathrm{CAT}(0)$ $2$–complex are intrinsically $\mathrm{CAT}(0)$

### Russell Ricks

Algebraic & Geometric Topology 21 (2021) 1723–1744
##### Abstract

Let $\kappa \le 0$, and let $X$ be a complete, locally finite $CAT\left(\kappa \right)$ polyhedral $2$–complex $X\phantom{\rule{-0.17em}{0ex}}$, 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.

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$\mathrm{CAT}(0)$, complex, subspaces