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 quotient, canonical, projection map associated with   refers to the following surjective map:

 
For any subset   (so in particular,   for every  ) the following holds:
 

The quotient space under   is the quotient set   equipped with the quotient topology, which is the topology whose open sets are the all those subsets   such that   is an open subset of   that is,   is open in the quotient topology on   if and only if   Thus,

 
Equivalently, the open sets of the quotient topology are the subsets of   that have an open preimage under the canonical map   (which is defined by  ). Similarly, a subset   is closed in   if and only if   is a closed subset of  

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) if it is surjective, and a subset   is open if and only if   is open. Equivalently, a surjection   is a quotient map if and only if for every subset     is closed in   if and only if   is closed in  

Final topology definition

Alternatively,   is a quotient map if it is onto and   is equipped with the final topology with respect to  

Saturated sets and quotient maps

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   (although   always holds for every subset   equality is in general not guaranteed; and a non-saturated set exists if and only if   is not injective). 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 or not the function   is a quotient map; non-saturated subsets are irrelevant to the definition of "quotient map" just as they are irrelevant to the open-set definition of continuity (because a function   is continuous if and only if for every saturated subset   of     being open in   implies   is open in  ).

Every quotient map is continuous but not every continuous map is a quotient map. A continuous surjection   fails to be a quotient map if and only if   has some saturated open subset   such that   is not open in   (this statement remains true if both instances of the word "open" are replaced with "closed").

Quotient space of fibers characterization

Given an equivalence relation   on   the canonical map   that sends   to its equivalence class   (that is,  ) is a quotient map that satisfies   for all  ; moreover, for all    

In fact, let   be a surjection between topological spaces (not yet assumed to be continuous or a quotient map) and declare for all   that   Then   is an equivalence relation on   such that for every     so that   is a singleton set, which thus induces a bijection   defined by   (this is well defined because   is a singleton set and   is just its unique element; that is,   for every  ). 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:

We say that   descends to the quotient. 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

Algebra

ReferencesEdit

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