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Standard function topology?

  1. Aug 24, 2009 #1
    Given two topological spaces X and Y, there is a standard way to define a topology on X x Y called the product topology.

    Given a subset S of X, there is a standard way to define a topology on S called the subset topology.

    Given an equivalence relation ~ on X, there is a standard way to define a topology on X/~ called the quotient topology.

    Are there any standard ways to define a topology various function spaces X->Y?
  2. jcsd
  3. Aug 24, 2009 #2
    The most common topology on the space of functions from X to Y is probably the product topology, which corresponds to the topology of point-wise convergence. One can think of the space of functions from X to Y, usually written [tex]Y^X[/tex], as a product of topological spaces: [tex]\prod_{x \in X} Y_x[/tex] where in this case [tex]Y_x = Y[/tex] for all x. The product topology on [tex]Y^X[/tex] is generated by a basis consisting of sets of the form [tex]\prod_{x \in X} U_x[/tex] where [tex]U_x[/tex] is open in Y for each x, and [tex]U_x = Y[/tex] for all but finitely many x. Interestingly, the topology on X does not play a role here at all, so X does not even need to be a topological space, and may simply be any set.

    The topology generated by basis elements of the form [tex]\prod_{x \in X} U_x[/tex], without the requirement that [tex]U_x = Y[/tex] for all but finitely many x, by the way, is called the box topology. The box topology is usually too fine to be of interest, which is why the product topology is usually preferred.

    In the case where Y is a metric space, one can define the topology of uniform convergence on [tex]Y^X[/tex], which is metrizable. The metric is given by [tex]d(f, g) = \min(\sup_{x \in X} d(f(x), g(x)), 1)[/tex].

    Hope this helps, and is at least somewhat understandable.
  4. Aug 24, 2009 #3
    You can look up "k-space", which is a class of topological spaces where there is a reasonable function-space topology. "Reasonable" can be explained as in category theory. That is, for the function space [itex]F(X,Y)[/itex] of functions from [itex]X[/itex] to [itex]Y[/itex] (whatever it is), there should be a natural identification between [itex]F(X \times Y, Z)[/itex] and [itex]F(X,F(Y,Z))[/itex] .
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