A complete metric space without non-trivial separable Lipschitz retracts

TL;DR

This paper constructs a complete metric space with the property that no non-trivial separable closed subset can be a Lipschitz retract, illustrating a unique geometric property in metric space theory.

Contribution

It introduces a novel complete metric space demonstrating the absence of non-trivial separable Lipschitz retracts, extending classical Banach space examples to metric spaces.

Findings

Constructed a complete metric space with continuum cardinality

Proved that all non-singleton closed separable subsets are not Lipschitz retracts

Provides a metric analogue to Banach space examples of non-complemented subspaces

Abstract

We construct a complete metric space $M$ of cardinality continuum such that every non-singleton closed separable subset of $M$ fails to be a Lipschitz retract of $M$. This provides a metric analogue to the various classical and recent examples of Banach spaces failing to have linearly complemented subspaces of prescribed smaller density character.