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 sequentially quotient maps, sequence coverings, 1-sequence coverings, and 2-sequence coverings. 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).
A subset of is said to be sequentially open in if whenever a sequence in converges (in ) to some point that belongs to then that sequence is necessarily eventually 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, is called a sequential space if Given a sequence in and a point in if and only if in Moreover, is the finest topology on for which this characterization of sequence convergence in holds.
A map is called sequentially continuous if is continuous, which happens if and only if for every sequence in and every if in then necessarily in Every continuous map is sequentially continuous although in general, the converse may fail to hold. In fact, a space is a sequential space if and only if it has the following universal property for sequential spaces:
- for every topological space and every map the map is continuous if and only if it is sequentially continuous.
The sequential closure in of a subset is the set consisting of all for which there exists a sequence in that converges to in A subset is called sequentially closed in if which happens if and only if whenever a sequence in converges in to some point then necessarily A subset is sequentially open (resp. sequentially closed) if and only if its complement is sequentially closed (resp. sequentially open). The space is called a Fréchet–Urysohn space if for every subset which happens if and only if every subspace of is a sequential space. Every first-countable space is a Fréchet–Urysohn space and thus also a sequential space. All pseudometrizable spaces, metrizable spaces, and second-countable spaces are first-countable.
A sequence in a set is by definition a function whose value at is denoted by (although the usual notation used with functions, such as parentheses or composition might be used in certain situations to improve readability). Statements such as "the sequence is injective" or "the image (i.e. range) of a sequence is infinite" as well as other terminology and notation that is defined for functions can thus be applied to sequences. A sequence is said to be a subsequence of another sequence if there exists a strictly increasing map (possibly denoted by instead) such that for every where this condition can be expressed in terms of function composition as: As usual, if is declared to be (such as by definition) a subsequence of then it should immediately be assumed that is strictly increasing. The notation and mean that the sequence is valued in the set
Throughout, let be a map between the topological spaces and  If is a sequence in then a sequence in is called a lift of by or an -lift of if (that is, for every ). The map is said to be able to lift a sequence if there exists a sequence such that If in addition the -lift can be chosen so that is a convergent sequence in then is called a convergent lift of by and is said to be able to lift to a convergent sequence.
The function is called a sequence covering if it can lift every convergent sequence in to a convergent sequence in It is called a 1-sequence covering if for every there exists some such that every sequence that converges to in has an -lift that converges to in It is a 2-sequence covering if is surjective and also for every and every every sequence that converges to in has an -lift that converges to in A map is a compact covering if for every compact there exists some compact subset such that
Sequentially quotient mappings
In analogy with the definition of sequential continuity, a map is called a sequentially quotient map if
is a quotient map, which happens if and only if for any subset is sequentially open if and only if this is true of in Sequentially quotient maps were introduced in Boone & Siwiec 1976 who defined them as above.
Every sequentially quotient map is necessarily surjective and sequentially continuous although they may fail to be continuous. If is a sequentially continuous surjection whose domain is a sequential space, then is a quotient map if and only if is a sequential space and is a sequentially quotient map.
Call a space sequentially Hausdorff if is a Hausdorff space, or equivalently, if every convergent sequence in converges in to at most one point. In an analogous manner, a "sequential version" of every other separation axiom can be defined in terms of whether or not the space possess it. Every Hausdorff space is necessarily sequentially Hausdorff. A sequential space is Hausdorff if and only if it is sequentially Hausdorff.
If is a sequentially continuous surjection then assuming that is sequentially Hausdorff, the following are equivalent:
- is sequentially quotient.
- Whenever is a convergent sequence in then there exists a convergent sequence in such that and is a subsequence of
- This can be restated as follows: Whenever is a convergent sequence in then there exists a subsequence with an -lift (i.e. is satisfied) such that converges in to some point
- Whenever is a convergent sequence in then there exists a convergent sequence in such that is a subsequence of
- This statement differs from (2) above only in that there are no requirements placed on the limits of the sequences (which becomes an important difference only when is not sequentially Hausdorff). This statement can be restated as follows: Whenever is a convergent sequence in then there exists a subsequence that has a convergent -lift to
- If is a continuous surjection onto a sequentially compact space then this condition holds even if is not sequentially Hausdorff.
If the assumption 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).
The map is called presequential if for every convergent sequence in such that is not eventually equal to the set is not sequentially closed in  where this set may also be described as:
Equivalently, is presequential if and only if for every convergent sequence in such that the set is not sequentially closed in
A surjective map between Hausdorff spaces is sequentially quotient if and only if it is sequentially continuous and a presequential map.
Related maps and sets
The preimage of a singleton set is called a fiber where if is a map and then the set is called the fiber of over . A map is called surjective (resp. injective, monotone) if each of its fibers is a non-empty set (resp. a subset of a singleton set, a connected subspace). A perfect map is a continuous closed surjection, each of whose fibers is compact.
A map is called hereditarily quotient if for every subset the restriction is a quotient map.
A map is called almost open if it is surjective and for every there exists some such that is a point of openness for which by definition means that for every open neighborhood of is a neighborhood of in If is an almost open map then the following equivalence holds: is an open map if and only if whenever belong to the same fiber of then for every open neighborhood of there exists some open neighborhood of such that Observe that unlike the left hand side, the right hand side of this characterization does not depend in any way on 's topology
- is almost open if and only if it is a 1-sequence covering.
- is an open map if and only if it is a 2-sequence covering.
- If is a compact covering map then is a quotient map.
- The following are equivalent:
- is a quotient map.
- is a sequentially quotient map.
- is a sequence covering.
- is a pseudo-open map.
- A map is called pseudo-open if for every and every open neighborhood of (i.e. an open subset of such that ), necessarily belongs to the interior (taken in ) of
- is a hereditarily quotient.
Suppose that is a continuous surjection from a regular space onto a Hausdorff space If the restriction is sequentially quotient for every open subset of then maps open subsets of to sequentially open subsets of Consequently, if is a continuous surjection between two sequential spaces where is also a regular space and is Hausdorff, then is an open map if and only if is sequentially quotient for every open subset of
A family of subsets of a topological space is said to be locally finite at a point if there exists some open neighborhood of such that the set is finite. Suppose is a continuous map between two Hausdorff first-countable spaces and let If there exists a sequence in such that (1) and (2) there exists some such that the family is not locally finite at then necessarily The converse is true if there does not exist any non-empty open subset of on which is constant.
Suppose is a continuous open surjection from a first-countable space onto a Hausdorff space and let be any non-empty subset, and let where denotes the closure of in Then given any and any sequence in that converges to there necessarily exists a sequence in that converges to as well as a subsequence of such that (that is, for all ). In short, this states that given a convergent sequence such that then for any other belonging to the same fiber as it is always possible to find a subsequence such that can be "lifted" by to a sequence that converges to
If is a sequence covering from a Hausdorff sequential space onto a Hausdorff first-countable space and if is such that the fiber is a countable set, then there exists some such that is a point of openness for Consequently, if is quotient map between two Hausdorff first-countable spaces and if every fiber of is countable, then is an almost open map and consequently, also a 1-sequence covering.
- Fréchet–Urysohn space
- Open map
- Perfect map – A continuous closed surjective map, each of whose fibers are also compact sets.
- Proper map – A map between topological spaces with the property that the preimage of every compact is compact
- Sequential space – A topological space that is can be characterized in terms of sequences
- Sequentially compact space
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