Sequence covering map
In mathematics, specifically topology, a sequence covering map is any of a class of maps between topological spaces whose definitions all somehow relate sequences in the codomain with sequences in the domain. Examples include maps, , , and . These classes of maps are closely related to sequential spaces. If the domain and/or codomain have certain additional topological properties (often, the spaces being Hausdorff and first-countable is more than enough) then these definitions become equivalent to other well-known classes of maps, such as open maps or quotient maps, for example. In these situations, characterizations of such properties in terms of convergent sequences might provide benefits similar to those provided by, say for instance, the characterization of continuity in terms of sequential continuity or the characterization of compactness in terms of sequential compactness (whenever such characterizations hold). Definitions Preliminaries A subset of is said to be if whenever a sequence in converges (in ) to some point that belongs to then that sequence is necessarily in (i.e. at most finitely many points in the sequence do not belong to ). The set of all sequentially open subsets of forms a topology on that is finer than 's given topology By definition,