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.