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,  is called a  if  
Given a sequence  in  and a point   in  if and only if  in  Moreover,  is the  topology on  for which this characterization of sequence convergence in  holds. 

A map  is called  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 : 
for every topological space  and every map  the map  is continuous if and only if it is sequentially continuous. 

The  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  in  if  which happens if and only if whenever a sequence in  converges in  to some point  then necessarily  
The space  is called a  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.

Sequence coverings

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  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  

The function  is called a  if for every convergent sequence  in   there exists a sequence  such that  
It is called a  if for every  there exists some  such that every sequence  that converges to  in  there exists a sequence  such that  and  converges to  in  
It is a  if  is surjective and also for every  and every  every sequence  and  converges to  in  there exists a sequence  such that  and  converges to  in  
A map  is a  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  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  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   if  is a Hausdorff space. 
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 
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). 
 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  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  is called  if for every convergent sequence  in  such that  is not eventually equal to  the set  is  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  sequentially closed in 

A surjective map  between Hausdorff spaces is sequentially quotient if and only if it is sequentially continuous and a presequential map.

Characterizations

If  is a continuous surjection between two first-countable Hausdorff spaces then the following statements are true: 
 is almost open if and only if it is a 1-sequence covering.
 An  is surjective map  with the property that for every  there exists some  such that  is a  for  which by definition means that for every open neighborhood  of   is a neighborhood of  in 
 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  if for every  and every open neighborhood  of  (meaning an open subset  such that ),  necessarily belongs to the interior (taken in ) of 
and if in addition both  and  are separable metric spaces then to this list may be appended:
 is a hereditarily quotient map.

Properties

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. Assume 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  and  are also sequential spaces, then  is an open map if and only if  is sequentially quotient (or equivalently, quotient) for every open subset  of  

Given an element  in the codomain of a (not necessarily surjective) continuous function  the following gives a sufficient condition for  to belong to 's image:  A family  of subsets of a topological space  is said to be  at a point  if there exists some open neighborhood  of  such that the set  is finite. 
Assume that  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  is  locally finite at  then  
The converse is true if there is no point at which  is locally constant; that is, if there does not exist any non-empty open subset of  on which  restricts to a constant map.

Sufficient conditions

Suppose  is a continuous open surjection from a first-countable space  onto a Hausdorff space  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 exists a sequence  in  that converges to  as well as a subsequence  of  such that  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 

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. 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.

See also

Notes

Citations

References

  
  
  
  
  
  
  
  
  
  
  
  
  
  

Topological graph theory