Local decomposition
local cut
A closed set in the space
is a local cut if there are a point
(a point at which the set
cuts the space) and a positive number
such that for any number
there is in the open set
, where
is the (open) ball of radius
with centre at
, a pair of points with the following property: Any continuum lying in
and containing this pair of points has a non-empty intersection with
. K. Menger and P.S. Urysohn proved that a closed set
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
at least).
A similar characterization of closed -dimensional sets in the
-dimensional space
was given by P.S. Aleksandrov (see Local linking).
Local decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_decomposition&oldid=43442