Quotient Metric Spaces
If M is a metric space with metric d, and ~ is an equivalence relation on M, then we can endow the quotient set M/~ with the following (pseudo)metric. Given two equivalence classes and, we define
where the infimum is taken over all finite sequences and with, . In general this will only define a pseudometric, i.e. does not necessarily imply that . However for nice equivalence relations (e.g., those given by gluing together polyhedra along faces), it is a metric. Moreover if M is a compact space, then the induced topology on M/~ is the quotient topology.
The quotient metric d is characterized by the following universal property. If is a metric map between metric spaces (that is, for all x, y) satisfying f(x)=f(y) whenever then the induced function, given by, is a metric map
A topological space is sequential if and only if it is a quotient of a metric space.
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