# Quotient topology

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## Definition

### Quotient topology by an equivalence relation

Suppose  is a topological space and  is an equivalence relation on . In other words,  partitions  into disjoint subsets, namely the equivalence classes under it. The quotient space of  by , or the quotient topology of  by , denoted , is defined as follows:

• As a set, it is the set of equivalence classes under .
• The topology on it is defined as the finest topology possible so that the quotient map , that sends every element  to its equivalence class, is a continuous map. Equivalently, the open sets in the topology on  are those subsets of  whose inverse image in  (which is the union of all the corresponding equivalence classes) is an open subset of .

The map  is a quotient map. In fact, the notion of quotient topology is equivalent to the notion of quotient map (somewhat similar to the first isomorphism theorem in group theory?)

### Quotient topology by a subset

Suppose  is a topological space and  is a subset of . The quotient space  is defined as the quotient space , where  is the equivalence relation that identifies all points of  with each other but not with any point outside , and does not identify any distinct points outside . In other words, all points of  become one equivalence class, and each single point outside  forms its own equivalence class.

Note that a notation of the form  should be interpreted carefully. In case  is a topological group and  is a subgroup, this notation is to be intepreted as the coset space, and not in terms of the description given above. Context is extremely important.

### Quotient topology by a subset with based topological space interpretation

Suppose  is a topological space and  is a subset of . We may be interested in the pair of topological spaces . In this context,  (as defined above) is often viewed as a based topological space, with basepoint chosen as the equivalence class of .