$L$-retracts

TL;DR

This paper investigates the concept of $L$-equivalence in locally convex spaces, exploring its connections with extension theorems and providing conditions and examples that highlight its properties and limitations.

Contribution

It introduces a new perspective on $L$-equivalence, relating it to extension theorems and demonstrating its non-invariance under certain mappings.

Findings

$L$-equivalence relates to extension properties of continuous functions.

Open or closed mapping properties are not preserved under $L$-equivalence.

Provides methods to construct examples of $L$-equivalent spaces and mappings.

Abstract

We study the relation of $L$-equivalence, which derives from the construction of the free locally convex spaces, through a concept that particularizes several notions related to the simultaneous extension of continuous functions. We also explore the relationship that this concept has with the Dugundji's extension theorem, and, based on this theorem we give sufficient conditions that allow us to identify these sets in different types of topological spaces. In particular, we present a method for constructing examples of $L$-equivalent mappings (and hence $L$-equivalent spaces) that show that the properties of being an open or closed mapping are not $L$-invariant.