# Neighboring mapping points theorem

## TL;DR

This paper introduces a new family of topological theorems extending Borsuk-Ulam and Radon theorems by replacing 'large to singleton' conditions with 'large to small' conditions, applicable to broader spaces.

## Contribution

It proposes a novel extension framework for classical topological theorems, broadening their applicability to new classes of spaces and mappings.

## Key findings

- Extended theorems cover mappings from m-sphere to n-space with m<n
- Generalized conditions from 'large to singleton' to 'large to small'
- Applicable to wider classes of topological spaces

## Abstract

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.