Free locally convex spaces and $L$-retracts

We study the relation of $L$-equivalence defined between Tychonoff spaces, that is, we study the topological isomorphisms of their respective free locally convex spaces. We introduce the concept of an $L$-retract in a Tychonoff space in terms of the existence of a special kind of simultaneous extensions of continuous functions, explore the relation of this concept with the Dugundji extension theorem, and find some conditions that allow us to identify $L$-retracts in various classes of topological spaces. As applications, we present a method for constructing examples of $L$-equivalent mappings and $L$-equivalent spaces and in particular, we show that the properties of being an open mapping or a perfect mapping are not $L$-invariant.