Local decomposition

local cut

A closed set $\Phi$ in the space $\mathbf R^n$ is a local cut if there are a point $a$ (a point at which the set $\Phi$ cuts the space) and a positive number $\epsilon$ such that for any number $\delta>0$ there is in the open set $O(a,\delta)\setminus\Phi$, where $O(a,\delta)$ is the (open) ball of radius $\delta$ with centre at $a$, a pair of points with the following property: Any continuum lying in $O(a,\epsilon)$ and containing this pair of points has a non-empty intersection with $\Phi$. K. Menger and P.S. Urysohn proved that a closed set $\Phi$ lying in a plane has dimension 1 if and only if it does not contain interior (with respect to the plane) points and locally cuts the plane (at one point $a$ at least).

A similar characterization of closed $(n-1)$-dimensional sets in the $n$-dimensional space $\mathbf R^n$ was given by P.S. Aleksandrov (see Local linking).