View source for Quotient mapping

A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q0768901.png" /> of a [[Topological space|topological space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q0768902.png" /> onto a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q0768903.png" /> for which a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q0768904.png" /> is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q0768905.png" /> if and only if its pre-image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q0768906.png" /> is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q0768907.png" />. If one is given a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q0768908.png" /> of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q0768909.png" /> onto a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689010.png" />, then there is on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689011.png" /> a strongest topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689012.png" /> (that is, one containing the greatest number of open sets) among all the topologies relative to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689013.png" /> is continuous. The topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689014.png" /> consists of all sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689016.png" /> is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689017.png" />. This topology is the unique topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689018.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689019.png" /> is a quotient mapping. Therefore <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689020.png" /> is called the quotient topology corresponding to the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689021.png" /> and the given topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689022.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689023.png" />.

The construction described above arises in studying decompositions of topological spaces and leads to an important operation — passing from a given topological space to a new one — a decomposition space. Suppose one is given a decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689024.png" /> of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689025.png" />, that is, a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689026.png" /> of non-empty pairwise-disjoint subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689027.png" /> that covers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689028.png" />. Then a projection mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689029.png" /> is defined by the rule: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689030.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689031.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689032.png" /> is now endowed with the quotient topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689033.png" /> corresponding to the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689034.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689035.png" /> and the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689036.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689037.png" /> is called a decomposition space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689038.png" />. Thus, up to a homeomorphism a circle can be represented as a decomposition space of a line segment, a sphere as a decomposition space of a disc, the Möbius band as a decomposition space of a rectangle, the projective plane as a decomposition space of a sphere, etc.

The following properties of quotient mappings, connected with considering diagrams, are important: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689039.png" /> be a continuous mapping with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689040.png" />. Then there are a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689041.png" />, a quotient mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689042.png" /> and a continuous one-to-one mapping (that is, a contraction) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689043.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689044.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689045.png" /> one can take the decomposition space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689046.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689047.png" /> into the complete pre-images of points under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689048.png" />, and the role of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689049.png" /> is then played by the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689050.png" />. Suppose one is given a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689051.png" /> and a quotient mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689052.png" />, where the following condition is satisfied: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689054.png" />, then also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689055.png" />. Then the unique mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689056.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689057.png" /> turns out to be continuous. The restriction of a quotient mapping to a subspace need not be a quotient mapping — even if this subspace is both open and closed in the original space. The Cartesian product of a quotient mapping and the identity mapping need not be a quotient mapping, nor need the Cartesian square of a quotient mapping be such. The restriction of a quotient mapping to a complete pre-image does not have to be a quotient mapping. More precisely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689058.png" /> is a quotient mapping and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689061.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689062.png" /> need not be a quotient mapping. This cannot occur if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689063.png" /> is open or closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689064.png" />.

These facts show that one must treat quotient mappings with care and that from the point of view of category theory the class of quotient mappings is not as harmonious and convenient as that of the continuous mappings, perfect mappings and open mappings (cf. [[Continuous mapping|Continuous mapping]]; [[Perfect mapping|Perfect mapping]]; [[Open mapping|Open mapping]]). However, the consideration of decomposition spaces and the  "diagram"  properties of quotient mappings mentioned above assure the class of quotient mappings of a position as one of the most important classes of mappings in topology. This class contains all surjective, continuous, open or closed mappings (cf. [[Closed mapping|Closed mapping]]). Quotient mappings play a vital role in the classification of spaces by the method of mappings. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689066.png" />-spaces are characterized as quotient spaces (that is, images under quotient mappings) of locally compact Hausdorff spaces, and sequential spaces are precisely the quotient spaces of metric spaces.

The majority of topological properties are not preserved under quotient mappings. Thus, a quotient space of a metric space need not be a Hausdorff space, and a quotient space of a separable metric space need not have a countable base. Therefore the question of the behaviour of topological properties under quotient mappings usually arises under additional restrictions on the pre-images of points or on the image space. It is known, for example, that if a compactum is homeomorphic to a decomposition space of a separable metric space, then the compactum is metrizable. Under a quotient mapping of a separable metric space on a regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076890/q07689067.png" />-space with the first axiom of countability, the image is metrizable. But there are topological invariants that are stable relative to any quotient mapping. These include, for example, sequentiality and an upper bound on tightness. In topological algebra quotient mappings that are at the same time algebra homeomorphisms often have much more structure than in general topology. Thus, an algebraic homomorphism of one topological group onto another that is a quotient mapping is necessarily an open mapping. Thanks to this, the range of topological properties preserved by quotient homomorphisms is rather broad (it includes, for example, metrizability).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Arkhangel'skii,   V.I. Ponomarev,   "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. General topology" , Addison-Wesley  (1966)  (Translated from French)</TD></TR></table>



====Comments====
The decomposition space is also called the quotient space.

Properties preserved by quotient mappings (or by open mappings, bi-quotient mappings, etc.) are surveyed in [[#References|[a2]]]. However, every topological space is an open quotient of a paracompact [[Regular space|regular space]], [[#References|[a1]]] (cf. also [[Paracompact space|Paracompact space]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Isbell,   "A note on complete closure algebras"  ''Math. Systems Theory'' , '''3'''  (1969)  pp. 310–312</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.A. Michael,   "A quintuple quotient quest"  ''Gen. Topol. Appl.'' , '''2'''  (1972)  pp. 91–138</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Engelking,   "General topology" , Heldermann  (1989)</TD></TR></table>