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 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.

Definitions

Preliminaries

A subset of is said to be ' if whenever a sequence in converges to some point that belongs to then that sequence is necessarily in . 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 :
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 .
Statements such as "the sequence is injective" or "the image 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 such that for every where this condition can be expressed in terms of function composition as:
As usual, if is declared to be 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:

  1. is sequentially quotient.
  2. Whenever is a convergent sequence in then there exists a convergent sequence in such that and is a subsequence of
  3. Whenever is a convergent sequence in then there exists a convergent sequence in such that is a subsequence of
    • This statement differs from above only in that there are no requirements placed on the limits of the sequences.
    • 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 would still imply the other two statement but the above characterization would no longer be guaranteed to hold.
This remains true even if the sequential continuity requirement on was strengthened to require continuity.
Instead of using the original definition, some authors define "sequentially quotient map" to mean a surjection that satisfies condition or alternatively, condition. If the codomain is sequentially Hausdorff then these definitions differs from the original in the added requirement of 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:

    1. is a quotient map.
    2. is a sequentially quotient map.
    3. is a sequence covering.
    4. is a pseudo-open map.
      • A map is called if for every and every open neighborhood of , necessarily belongs to the interior of
    and if in addition both and are separable metric spaces then to this list may be appended:

    1. 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 for every open subset of
Given an element in the codomain of a 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 and 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.