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 formula_1 of formula_2 is said to be if whenever a sequence in formula_3 converges (in formula_2) to some point that belongs to formula_5 then that sequence is necessarily in formula_1 (i.e. at most finitely many points in the sequence do not belong to formula_1). The set formula_8 of all sequentially open subsets of formula_2 forms a topology on formula_3 that is finer than formula_3's given topology formula_12 
By definition, formula_2 is called a if formula_14 
Given a sequence formula_15 in formula_3 and a point formula_17 formula_18 in formula_2 if and only if formula_18 in formula_21 Moreover, formula_8 is the topology on formula_3 for which this characterization of sequence convergence in formula_2 holds. 
A map formula_25 is called if formula_26 is continuous, which happens if and only if for every sequence formula_27 in formula_3 and every formula_17 if formula_18 in formula_2 then necessarily formula_32 in formula_33 
Every continuous map is sequentially continuous although in general, the converse may fail to hold. 
In fact, a space formula_2 is a sequential space if and only if it has the following : 
The in formula_2 of a subset formula_39 is the set formula_40 consisting of all formula_41 for which there exists a sequence in formula_1 that converges to formula_43 in formula_44 
A subset formula_39 is called in formula_2 if formula_47 which happens if and only if whenever a sequence in formula_1 converges in formula_2 to some point formula_41 then necessarily formula_51 A subset formula_39 is sequentially open (resp. sequentially closed) if and only if its complement formula_53 is sequentially closed (resp. sequentially open). 
The space formula_2 is called a if formula_55 for every subset formula_56 which happens if and only if every subspace of formula_2 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.
Lifting sequences.
A sequence formula_58 in a set formula_3 is by definition a function formula_60 whose value at formula_61 is denoted by formula_62 (although the usual notation used with functions, such as parentheses formula_63 or composition formula_64 might be used in certain situations to improve readability). 
Statements such as "the sequence formula_15 is injective" or "the image (i.e. range) formula_66 of a sequence formula_15 is infinite" as well as other terminology and notation that is defined for functions can thus be applied to sequences. 
A sequence formula_68 is said to be a of another sequence formula_15 if there exists a strictly increasing map formula_70 (possibly denoted by formula_71 instead) such that formula_72 for every formula_73 where this condition can be expressed in terms of function composition formula_74 as: formula_75 
As usual, if formula_76 is declared to be (such as by definition) a subsequence of formula_15 then it should immediately be assumed that formula_70 is strictly increasing. 
The notation formula_79 and formula_80 mean that the sequence formula_15 is valued in the set formula_82 
Throughout, let formula_83 be a map between the topological spaces formula_2 and formula_33 
If formula_86 is a sequence in formula_87 then a sequence formula_88 in formula_3 is called a ' of formula_90 by formula_91 or an ' if formula_92 (that is, formula_93 for every formula_61). 
The map formula_91 is said to be a sequence formula_90 if there exists a sequence formula_88 such that formula_98 
If in addition the formula_91-lift formula_15 can be chosen so that formula_15 is a convergent sequence in formula_2 then formula_15 is called a ' of formula_90 by formula_91 and formula_91 is said to be '. 
The function formula_83 is called a if it can lift every convergent sequence formula_90 in formula_87 to a convergent sequence in formula_110 
It is called a if for every formula_111 there exists some formula_112 such that every sequence formula_86 that converges to formula_114 in formula_35 has an formula_91-lift that converges to formula_43 in formula_44 
It is a if formula_83 is surjective and also for every formula_111 and every formula_121 every sequence formula_86 that converges to formula_114 in formula_35 has an formula_91-lift that converges to formula_43 in formula_44 
A map formula_83 is a if for every compact formula_129 there exists some compact subset formula_130 such that formula_131
Sequentially quotient mappings.
In analogy with the definition of sequential continuity, a map formula_25 is called a if 
is a quotient map, which happens if and only if for any subset formula_134 formula_1 is sequentially open formula_35 if and only if this is true of formula_137 in formula_44 
Sequentially quotient maps were introduced in who defined them as above. 
Every sequentially quotient map is necessarily surjective and sequentially continuous although they may fail to be continuous. 
If formula_25 is a sequentially continuous surjection whose domain formula_2 is a sequential space, then formula_25 is a quotient map if and only if formula_35 is a sequential space and formula_25 is a sequentially quotient map. 
Call a space formula_35 if formula_145 is a Hausdorff space, or equivalently, if every convergent sequence in formula_35 converges in formula_35 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 formula_145 possess it. 
Every Hausdorff space is necessarily sequentially Hausdorff. A sequential space is Hausdorff if and only if it is sequentially Hausdorff. 
If formula_25 is a sequentially continuous surjection then assuming that formula_35 is sequentially Hausdorff, the following are equivalent:
If the assumption that formula_87 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 formula_83 was strengthened to require (ordinary) continuity. 
Instead of using the original definition, some authors define "sequentially quotient map" to mean a surjection that satisfies condition (2) or alternatively, condition (3). If the codomain is sequentially Hausdorff then these definitions differs from the original in the added requirement of continuity (rather than merely requiring sequential continuity).
The map formula_25 is called if for every convergent sequence formula_154 in formula_35 such that formula_90 is not eventually equal to formula_157 the set formula_158 is sequentially closed in formula_159 where this set may also be described as:
Equivalently, formula_25 is presequential if and only if for every convergent sequence formula_154 in formula_35 such that formula_164 the set formula_165 is sequentially closed in formula_44
A surjective map formula_25 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 ' where if formula_83 is a map and formula_169 then the set formula_170 is called the '. 
A map is called ' (resp. ', ') if each of its fibers is a non-empty set (resp. a subset of a singleton set, a connected subspace). A ' is a continuous closed surjection, each of whose fibers is compact. 
A map formula_83 is called if for every subset formula_134 the restriction formula_173 is a quotient map.
A map formula_83 is called ' if it is surjective and for every formula_111 there exists some formula_112 such that formula_43 is a ' for formula_178 which by definition means that for every open neighborhood formula_179 of formula_180 formula_181 is a neighborhood of formula_182 in formula_183 
If formula_25 is an almost open map then the following equivalence holds: formula_83 is an open map if and only if whenever formula_186 belong to the same fiber of formula_91 then for every open neighborhood formula_188 of formula_189 there exists some open neighborhood formula_190 of formula_191 such that formula_192 Observe that unlike the left hand side, the right hand side of this characterization does not depend in any way on formula_87's topology formula_194
Characterizations.
If formula_25 is a continuous surjection between two first-countable Hausdorff spaces then the following statements are true: 
Properties.
Suppose that formula_83 is a continuous surjection from a regular space formula_3 onto a Hausdorff space formula_183 If the restriction formula_199 is sequentially quotient for every open subset formula_179 of formula_3 then formula_83 maps open subsets of formula_3 to sequentially open subsets of formula_183 
Consequently, if formula_83 is a continuous surjection between two sequential spaces where formula_3 is also a regular space and formula_87 is Hausdorff, then formula_83 is an open map if and only if formula_199 is sequentially quotient for every open subset formula_179 of formula_110 
A family formula_212 of subsets of a topological space formula_2 is said to be at a point formula_41 if there exists some open neighborhood formula_179 of formula_43 such that the set formula_217 is finite. 
Suppose formula_83 is a continuous map between two Hausdorff first-countable spaces and let formula_219 
If there exists a sequence formula_220 in formula_87 such that (1) formula_154 and (2) there exists some formula_41 such that the family formula_224 is locally finite at formula_180 then necessarily formula_226 
The converse is true if there does not exist any non-empty open subset of formula_3 on which formula_91 is constant.
Sufficient conditions.
Suppose formula_83 is a continuous open surjection from a first-countable space formula_3 onto a Hausdorff space formula_87 and let formula_232 be any non-empty subset, and let formula_233 where formula_234 denotes the closure of formula_235 in formula_183 
Then given any formula_237 and any sequence formula_15 in formula_239 that converges to formula_180 there necessarily exists a sequence formula_241 in formula_239 that converges to formula_243 as well as a subsequence formula_244 of formula_15 such that formula_246 (that is, formula_247 for all formula_248). 
In short, this states that given a convergent sequence formula_249 such that formula_18 then for any other formula_251 belonging to the same fiber as formula_180 it is always possible to find a subsequence formula_244 such that formula_254 can be "lifted" by formula_91 to a sequence that converges to formula_256
If formula_83 is a sequence covering from a Hausdorff sequential space formula_3 onto a Hausdorff first-countable space formula_87 and if formula_111 is such that the fiber formula_170 is a countable set, then there exists some formula_112 such that formula_43 is a point of openness for formula_264 
Consequently, if formula_83 is quotient map between two Hausdorff first-countable spaces and if every fiber of formula_91 is countable, then formula_83 is an almost open map and consequently, also a 1-sequence covering.