  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.
