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 .