quotient space

English

Noun

quotient space (plural quotient spaces)

1. (topology and algebra) A space obtained from another by identification of points that are equivalent to one another in some equivalence relation.
   • 1983, K. D. Joshi, Introduction to General Topology, New Age International, page 129:

     Thus if ${\mathfrak {D}}$ is an arbitrary decomposition of a space $X$ into mutually disjoint subsets, then the corresponding quotient space is obtained by 'shrinking' or 'identifying' each member of ${\mathfrak {D}}$ to a single point. For this reason, the quotient spaces are sometimes called identification spaces and quotient maps as^[sic] identification maps.

   • 1989, unnamed translator, Nicolas Bourbaki, Elements of Mathematics: General Topology: Chapters 1-4, [1971, N. Bourbaki, Éléments de Mathématique: Topologie Générale 1-4, Masson], Springer, page 110,

     PROPOSITION 6. Every quotient space of a connected space is connected.

   • 2004, Silvia Biasotti, Bianca Falcidieno, Michela Spagnuolo, “6: Surface Shape Understanding Based on Extended Reeb Graphs”, in Sanjay Rana, editor, Topological Data Structures for Surfaces: An Introduction to Geographical Information Science, Wiley, page 96:

     The quotient space obtained from such a relation is called extended Reeb (ER) quotient space. Moreover, the ER quotient space, which is an abstract sub-space of M* and is independent from the geometry, may be represented as a traditional graph, which is called the extended Reeb graph (ERG).

Synonyms

• (space composed of points equivalent to each other in some relation): identification space

Related terms

• quotient map
• quotient topology

See also

• equivalence class
• quotient group (group theory)

Further reading

• Equivalence class on Wikipedia.Wikipedia
• Quotient space (topology) on Wikipedia.Wikipedia