Let \(S^n\) be the n-sphere with the geodesic metric and of diameter \(\pi \) . The intrinsic Čech complex \(\check{\textrm{C}}(S^n;r)\) is the nerve of all open balls of radius r in \(S^n\) . In this paper, we show how to control the homotopy connectivity of \(\check{\textrm{C}}(S^n;r)\) in terms of coverings of spheres. Our upper bound on the connectivity, which is sharp in the case \(n=1\) , comes from the chromatic numbers of Borsuk graphs of spheres. Our lower bound is obtained using the conicity (in the sense of Barmak) of \(\check{\textrm{C}}(X;r)\) for sufficiently dense, finite subsets X of \(S^n\) . Our bounds imply the new result that for \(n\ge 1\) , the homotopy type of \(\check{\textrm{C}}(S^n;r)\) changes infinitely many times as r varies over \((0,\pi )\) ; we conjecture only countably many times. Additionally, we lower bound the homological dimension of Čech complexes of finite subsets X of \(S^n\) in terms of packings of X.
