In this thesis we prove the solvability of the satisfiability problem for various classes of unquantified set-theoretical formulae. In particular, we will provide satisfiability tests that given a formula as input produce a model for it, if any exists. We will also show how the decidability of certain fragments of set theory can be used to prove the solvability of the satisfiability problem for some unquantified languages involving topological notions. In particular, a list of topological statements whose validity can be checked by our algorithms is given. The underlying motivation for this study is to enrich the class of theoretical results that can be used for a set-theoretic proof verifier; we also provide lower bounds for what is undecidable in set theory and topology.