The main goal of this talk is to extend results from point-free topology to more general settings. We start by studying how the notions of sublocale and localic map can be extended from locales (i.e. complete Heyting algebras) to Heyting semilattices. Motivated by the case of Heyting semilattices, we introduce the category of basic zero-dimensional spaces. This category will provide a general framework for the study of continuity and openness, bringing together classical and point-free topology (as well as other areas). We then extend to this new environment two fundamental results from point-free topology. One about continuity, the fact that the localic preimage exists and is a coframe homomorphism; the other about openness, the Joyal-Tierney's Theorem (characterizing open localic maps as the localic maps whose left adjoint is a complete Heyting homomorphism).