Topology of pointwise convergence: Difference between revisions

Latest revision as of 16:25, 20 December 2010

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:

It is the natural topology such that convergence of a sequence of elements in the topology is equivalent to their pointwise convergence as functions.
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 $X$ .

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 ==Definition==
-Suppose <math>X</math> and <math>Y</math> are [[topological space]]s. Let <math>C(X,Y)</math> denote the space of [[continuous map]]s from <math>X</math> to <math>Y</math>. The '''topology of pointwise convergence''' on <math>C(X,Y)</math> is a topology such that convergence of a sequence of elements in the topology is equivalent to their pointwise convergence as functions.
+Suppose <math>X</math> and <math>Y</math> are [[topological space]]s. Let <math>C(X,Y)</math> denote the space of [[continuous map]]s from <math>X</math> to <math>Y</math>. The '''topology of pointwise convergence''' on <math>C(X,Y)</math> is defined in the following equivalent ways:
-{{fillin}}
+# It is the natural topology such that convergence of a sequence of elements in the topology is equivalent to their pointwise convergence as functions.
+# It is the topology on <math>C(X,Y)</math> arising as the [[subspace topology]] from the [[product topology]] on the space of ''all'' functions <math>Y^X</math>.
+In particular, the ''topology'' of pointwise convergence is little influenced by the topology of <math>X</math>, although the underlying ''set'' of the topology is influenced by <math>X</math>.