Topology of pointwise convergence

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This article defines a function space topology i.e. a topology on the collection of continuous maps between two topological spaces

Definition

Suppose X and Y are topological spaces. Let C(X,Y) denote the space of continuous maps from X to Y. The topology of pointwise convergence on C(X,Y) is defined in the following equivalent ways:

  1. It is the natural topology such that convergence of a sequence of elements in the topology is equivalent to their pointwise convergence as functions.
  2. It is the topology on C(X,Y) arising as the subspace topology from the product topology on the space of all functions Y^X.

In particular, the topology of pointwise convergence is little influenced by the topology of X, although the underlying set of the topology is influenced by X.