Quotient map

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This article defines a property of continuous maps between topological spaces

Definition

Symbol-free definition

A continuous map between topological spaces is termed a quotient map if it is surjective, and if a set in the range space is open iff its inverse image is open in the domain space.

Definition with symbols

Let X,Y be topological spaces and f:X \to Y be continuous maps. f is termed a quotient map if it is sujective and if U \subset Y is open iff f^{-1}(U) is open in X.

Significance

Given a topological space X, a set Y and a surjective map f:X \to Y, we can prescribe a unique topology on Y, the so-called quotient topology, such that f is a quotient map. Moreover, this is the coarsest topology for which f becomes continuous.

Also, the study of a quotient map is equivalent to the study of the equivalence relation on X given by x \simeq y \iff f(x) = f(y).

Relation with other properties

Stronger properties