Quotient topology

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Definition

Quotient topology by an equivalence relation

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

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

The map q:X \to X/\sim 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 X is a topological space and A is a subset of X. The quotient space X/A is defined as the quotient space \! X/\sim, where \sim is the equivalence relation that identifies all points of A with each other but not with any point outside A, and does not identify any distinct points outside A. In other words, all points of A become one equivalence class, and each single point outside A forms its own equivalence class.

Note that a notation of the form X/A should be interpreted carefully. In case X is a topological group and A 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 X is a topological space and A is a subset of X. We may be interested in the pair of topological spaces (X,A). In this context, X/A (as defined above) is often viewed as a based topological space, with basepoint chosen as the equivalence class of A.