Urysohn and Tietze extensions of Lipschitz functions

TL;DR

This paper explores the extension of Lipschitz functions in metric spaces, generalizing classical theorems like Tietze and Urysohn to Lipschitz contexts, and establishing conditions for function separation.

Contribution

It introduces generalized Tietze-Lipschitz extensions and Urysohn-lemma for Lipschitz functions, expanding classical topological results to metric space functions.

Findings

Established conditions for Lipschitz function extension without increasing Lipschitz constant.

Generalized Tietze extension theorem for Lipschitz functions.

Provided necessary and sufficient conditions for Lipschitz functions to separate subsets.

Abstract

Let (X,d) be a metric space and $ \alpha > 0 $. In this paper, we study extensions of some complex-valued Lipschitz functions, from some special subset $ X_0 $ to X. These extensions are with no-increasing Lipschitz number or the smallest Lipschitz number. Moreover, we show that under some conditions, Tietze extension theorem can be generalized for Lipschitz functions and call it Tietze-Lipschitz extension. Furthermore, we generalize Urysohn-lemma for Lipschitz functions. In fact we present a necessary and sufficient condition for that Lipschitz functions separate subsets of X.