  We introduce and study a new family of extensions for the Borsuk-Ulam and
topological Radon type theorems. The defining idea for this new family is to
replace requirements of the form `a subset that is large in some sense goes to
a singleton' with requirements of the milder form `a subset that is large in
some sense goes to a subset that is small in some sense'. This approach covers
the case of mappings m-sphere to n-space with m<n and extends to wider classes
of spaces.
