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$L$-retracts

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.