Quotient space (topology)

In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes). In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space.

Illustration of the construction of a topological sphere as the quotient space of a disk, by gluing together to a single point the points (in blue) of the boundary of the disk.

Intuitively speaking, the points of each equivalence class are identified or "glued together" for forming a new topological space. For example, identifying the points of a sphere that belong to the same diameter produces the projective plane as a quotient space.

DefinitionEdit

Let   be a topological space, and let   be an equivalence relation on   The quotient set   is the set of equivalence classes of elements of   The equivalence class of   is denoted  

The construction of   defines a canonical surjection   As discussed below,   is a quotient mapping, commonly called the canonical quotient map, or canonical projection map, associated to  

The quotient space under   is the set   equipped with the quotient topology, whose open sets are those subsets   whose preimage   is open. In other words,   is open in the quotient topology on   if and only if   is open in   Similarly, a subset   is closed if and only if   is closed in  

The quotient topology is the final topology on the quotient set, with respect to the map  

Quotient mapEdit

A map   is a quotient map (sometimes called an identification map[1]) if it is surjective and   is equipped with the final topology induced by   The latter condition admits two more-elementary phrasings: a subset   is open (closed) if and only if   is open (resp. closed). Every quotient map is continuous but not every continuous map is a quotient map.

Saturated sets

A subset   of   is called saturated (with respect to  ) if it is of the form   for some set   which is true if and only if   The assignment   establishes a one-to-one correspondence (whose inverse is  ) between subsets   of   and saturated subsets of   With this terminology, a surjection   is a quotient map if and only if for every saturated subset   of     is open in   if and only if   is open in   In particular, open subsets of   that are not saturated have no impact on whether the function   is a quotient map (or, indeed, continuous: a function   is continuous if and only if, for every saturated   such that   is open in  , the set   is open in  ).

Indeed, if   is a topology on   and   is any map then set   of all   that are saturated subsets of   forms a topology on   If   is also a topological space then   is a quotient map (respectively, continuous) if and only if the same is true of  

Quotient space of fibers characterization

Given an equivalence relation   on   denote the equivalence class of a point   by   and let   denote the set of equivalence classes. The map   that sends points to their equivalence classes (that is, it is defined by   for every  ) is called the canonical map. It is a surjective map and for all     if and only if   consequently,   for all   In particular, this shows that the set of equivalence class   is exactly the set of fibers of the canonical map   If   is a topological space then giving   the quotient topology induced by   will make it into a quotient space and make   into a quotient map. Up to a homeomorphism, this construction is representative of all quotient spaces; the precise meaning of this is now explained.

Let   be a surjection between topological spaces (not yet assumed to be continuous or a quotient map) and declare for all   that   if and only if   Then   is an equivalence relation on   such that for every     which implies that   (defined by  ) is a singleton set; denote the unique element in   by   (so by definition,  ). The assignment   defines a bijection   between the fibers of   and points in   Define the map   as above (by  ) and give   the quotient topology induced by   (which makes   a quotient map). These maps are related by:

 
From this and the fact that   is a quotient map, it follows that   is continuous if and only if this is true of   Furthermore,   is a quotient map if and only if   is a homeomorphism (or equivalently, if and only if both   and its inverse are continuous).

Related definitionsEdit

A hereditarily quotient map is a surjective map   with the property that for every subset   the restriction   is also a quotient map. There exist quotient maps that are not hereditarily quotient.

ExamplesEdit

  • Gluing. Topologists talk of gluing points together. If   is a topological space, gluing the points   and   in   means considering the quotient space obtained from the equivalence relation   if and only if   or   (or  ).
  • Consider the unit square   and the equivalence relation ~ generated by the requirement that all boundary points be equivalent, thus identifying all boundary points to a single equivalence class. Then   is homeomorphic to the sphere  
 
For example,   is homeomorphic to the circle  
  • Adjunction space. More generally, suppose   is a space and   is a subspace of   One can identify all points in   to a single equivalence class and leave points outside of   equivalent only to themselves. The resulting quotient space is denoted   The 2-sphere is then homeomorphic to a closed disc with its boundary identified to a single point:  
  • Consider the set   of real numbers with the ordinary topology, and write   if and only if   is an integer. Then the quotient space   is homeomorphic to the unit circle   via the homeomorphism which sends the equivalence class of   to  
  • A generalization of the previous example is the following: Suppose a topological group   acts continuously on a space   One can form an equivalence relation on   by saying points are equivalent if and only if they lie in the same orbit. The quotient space under this relation is called the orbit space, denoted   In the previous example   acts on   by translation. The orbit space   is homeomorphic to  
    • Note: The notation   is somewhat ambiguous. If   is understood to be a group acting on   via addition, then the quotient is the circle. However, if   is thought of as a topological subspace of   (that is identified as a single point) then the quotient   (which is identifiable with the set  ) is a countably infinite bouquet of circles joined at a single point  
  • This next example shows that it is in general not true that if   is a quotient map then every convergent sequence (respectively, every convergent net) in   has a lift (by  ) to a convergent sequence (or convergent net) in   Let   and   Let   and let   be the quotient map   so that   and   for every   The map   defined by   is well-defined (because  ) and a homeomorphism. Let   and let   be any sequences (or more generally, any nets) valued in   such that   in   Then the sequence
     
    converges to   in   but there does not exist any convergent lift of this sequence by the quotient map   (that is, there is no sequence   in   that both converges to some   and satisfies   for every  ). This counterexample can be generalized to nets by letting   be any directed set, and making   into a net by declaring that for any     holds if and only if both (1)   and (2) if   then the  -indexed net defined by letting   equal   and equal to   has no lift (by  ) to a convergent  -indexed net in  

PropertiesEdit

Quotient maps   are characterized among surjective maps by the following property: if   is any topological space and   is any function, then   is continuous if and only if   is continuous.

The quotient space   together with the quotient map   is characterized by the following universal property: if   is a continuous map such that   implies   for all   then there exists a unique continuous map   such that   In other words, the following diagram commutes:

One says that   descends to the quotient for expressing this, that is that it factorizes through the quotient space. The continuous maps defined on   are, therefore, precisely those maps which arise from continuous maps defined on   that respect the equivalence relation (in the sense that they send equivalent elements to the same image). This criterion is copiously used when studying quotient spaces.

Given a continuous surjection   it is useful to have criteria by which one can determine if   is a quotient map. Two sufficient criteria are that   be open or closed. Note that these conditions are only sufficient, not necessary. It is easy to construct examples of quotient maps that are neither open nor closed. For topological groups, the quotient map is open.

Compatibility with other topological notionsEdit

Separation

  • In general, quotient spaces are ill-behaved with respect to separation axioms. The separation properties of   need not be inherited by   and   may have separation properties not shared by  
  •   is a T1 space if and only if every equivalence class of   is closed in  
  • If the quotient map is open, then   is a Hausdorff space if and only if ~ is a closed subset of the product space  

Connectedness

Compactness

  • If a space is compact, then so are all its quotient spaces.
  • A quotient space of a locally compact space need not be locally compact.

Dimension

See alsoEdit

Topology

  • Covering space – Type of continuous map in topology
  • Disjoint union (topology) – space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology
  • Final topology – Finest topology making some functions continuous
  • Mapping cone (topology) – topological construction
  • Product space – Topology on Cartesian products of topological spaces
  • Subspace (topology)
  • Topological space – Mathematical space with a notion of closeness

Algebra

NotesEdit

  1. ^ Brown 2006, p. 103.

ReferencesEdit