We mention some situations in probability where sequential continuity is much more natural than continuity and which lead to the notion of sequential envelope. We present a survey of constructions in the realm of sequential convergence which are analogous to well-known topological constructions. From the categorical viewpoint the constructions are very similar, but the properties of the resulting object are strikingly different. In particular, we compare the Čech--Stone compactification, the Hewitt realcompacti-fication, topological group and ring completions, and the Stone duality with their sequential counterparts. We study equivalence of the sequential convergence spaces of events with respect to probability measures. Finally, we deal with the relationship between an $s$-perfect field of sets and the induced 0-dimensional topological space. We show that the category of $N$-compact topological spaces is isomophic with the dual category of a subcategory of $s$-perfect fields of sets and sequentially continuous homomorphisms.