3 problems

For each dimension $n\ge 1$ and scale $r>0$, let $S^n$ be the unit $n$-sphere and let $\mathrm{\check{C}}(S^n;r)$ denote its Čech complex at scale $r$. Finite CW-complex conjecture…

For $n\ge 1$, let $S^n$ be the unit $n$-sphere and let $\mathrm{\check{C}}(S^n;r)$ denote its Čech complex at scale $r\in(0,\pi)$. Write $\operatorname{conn}(K)$ for the connectivi…

Steinhaus–Čech equivalence conjecture. The Čech filtration constructed from $X$ is isomorphic to the cover filtration on $X$ constructed from $\check{C}_R(X)$.