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.
[1][2][3][4] 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).
(although the usual notation used with functions, such as parentheses
is infinite" as well as other terminology and notation that is defined for functions can thus be applied to sequences.
if there exists a strictly increasing map
where this condition can be expressed in terms of function composition
In analogy with the definition of sequential continuity, a map
Sequentially quotient maps were introduced in Boone & Siwiec 1976 who defined them as above.
is a sequentially continuous surjection whose domain
[6] In an analogous manner, a "sequential version" of every other separation axiom can be defined in terms of whether or not the space
is a sequentially continuous surjection then assuming that
is sequentially Hausdorff were to be removed, then statement (2) would still imply the other two statement but the above characterization would no longer be guaranteed to hold (however, if points in the codomain were required to be sequentially closed then any sequentially quotient map would necessarily satisfy condition (3)).
This remains true even if the sequential continuity requirement on
was strengthened to require (ordinary) continuity.
Instead of using the original definition, some authors define "sequentially quotient map" to mean a continuous surjection that satisfies condition (2) or alternatively, condition (3).
If the codomain is sequentially Hausdorff then these definitions differs from the original only in the added requirement of continuity (rather than merely requiring sequential continuity).
is called presequential if for every convergent sequence
is a continuous surjection between two first-countable Hausdorff spaces then the following statements are true:[7][8][9][10][11][12][3][4] and if in addition both
are separable metric spaces then to this list may be appended: The following is a sufficient condition for a continuous surjection to be sequentially open, which with additional assumptions, results in a characterization of open maps.
is a continuous surjection from a regular space
in the codomain of a (not necessarily surjective) continuous function
is a continuous map between two Hausdorff first-countable spaces and let
is locally constant; that is, if there does not exist any non-empty open subset of
is a continuous open surjection from a first-countable space
In short, this states that given a convergent sequence
The following shows that under certain conditions, a map's fiber being a countable set is enough to guarantee the existence of a point of openness.
is a sequence covering from a Hausdorff sequential space
is quotient map between two Hausdorff first-countable spaces and if every fiber of