Topology of uniform convergence: Difference between revisions

Revision as of 01:13, 2 February 2008

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

Definition

Suppose $X$ is a topological space and $Y$ is a uniform space (for instance, $Y$ may be a metric space or a $T_{0}$ -topological group). Let $C (X, Y)$ denote the space of continuous functions from $X$ to $Y$ . Then, we can define a topology of uniform convergence on $C (X, Y)$ as follows

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-{{function space property}}
+{{function space topology}}
 ==Definition==
 Suppose <math>X</math> is a [[topological space]] and <math>Y</math> is a [[uniform space]] (for instance, <math>Y</math> may be a [[metric space]] or a <math>T_0</math>-[[topological group]]). Let <math>C(X,Y)</math> denote the space of continuous functions from <math>X</math> to <math>Y</math>. Then, we can define a '''topology of uniform convergence''' on <math>C(X,Y)</math> as follows