Properties
Quotient maps q : X → Y are characterized among surjective maps by the following property: if Z is any topological space and f : Y → Z is any function, then f is continuous if and only if f ∘ q is continuous.
The quotient space X/~ together with the quotient map q : X → X/~ is characterized by the following universal property: if g : X → Z is a continuous map such that a ~ b implies g(a) = g(b) for all a and b in X, then there exists a unique continuous map f : X/~ → Z such that g = f ∘ q. We say that g descends to the quotient.
The continuous maps defined on X/~ are therefore precisely those maps which arise from continuous maps defined on X that respect the equivalence relation (in the sense that they send equivalent elements to the same image). This criterion is constantly used when studying quotient spaces.
Given a continuous surjection f : X → Y it is useful to have criteria by which one can determine if f is a quotient map. Two sufficient criteria are that f be open or closed. Note that these conditions are only sufficient, not necessary. It is easy to construct examples of quotient maps that are neither open nor closed.
Read more about this topic: Quotient Space
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