  Within dependent type theory, we provide a topological counterpart of
well-founded trees (for short, W-types) by using a proof-relevant version of
the notion of inductively generated suplattices introduced in the context of
formal topology under the name of inductively generated basic covers. In more
detail, we show, firstly, that in Homotopy Type Theory, W-types and proof
relevant inductively generated basic covers are propositionally mutually
encodable. Secondly, we prove they are definitionally mutually encodable in the
Agda implementation of intensional Martin-Loef's type theory. Finally, we
reframe the equivalence in the Minimalist Foundation framework by introducing
well-founded predicates as the logical counterpart for predicates of dependent
W-types. All the results have been checked in the Agda proof-assistant.
