Compact-open topology: Difference between revisions

Revision as of 16:18, 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. The compact-open topology is a topology we can define on the space of continuous functions $C (X, Y)$ from $X$ to $Y$ as follows.

For a compact subset $K \subseteq X$ and an open subset $U \subseteq Y$ , we define $W (K, U)$ as the set of all continuous maps $f : X \to Y$ such that $f (K) \subseteq U$ . The compact-open topology is the topology with subbasis as the set of all $W (K, U)$ s.

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	Suppose <math>X</math> and <math>Y</math> are [[topological space]]s. The '''compact-open topology''' is a topology we can define on the space of continuous functions <math>C(X,Y)</math> from <math>X</math> to <math>Y</math> as follows.		Suppose <math>X</math> and <math>Y</math> are [[topological space]]s. The '''compact-open topology''' is a topology we can define on the space of continuous functions <math>C(X,Y)</math> from <math>X</math> to <math>Y</math> as follows.

	For a compact subset <math>K \~~subset~~ X</math> and an open subset <math>U \~~subset~~ Y</math>, we define <math>W(K,U)</math> as the set of all continuous maps <math>f:X \to Y</math> such that <math>f(K) \~~subset~~ U</math>. The compact-open topology is the topology with [[subbasis]] as the set of all <math>W(K,U)</math>s.		For a compact subset <math>K \subseteq X</math> and an open subset <math>U \subseteq Y</math>, we define <math>W(K,U)</math> as the set of all continuous maps <math>f:X \to Y</math> such that <math>f(K) \subseteq U</math>. The compact-open topology is the topology with [[subbasis]] as the set of all <math>W(K,U)</math>s.