# 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 ${\displaystyle {\mathfrak {D}}}$ is an arbitrary decomposition of a space ${\displaystyle X}$ into mutually disjoint subsets, then the corresponding quotient space is obtained by 'shrinking' or 'identifying' each member of ${\displaystyle {\mathfrak {D}}}$ 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, 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).