Compact-open topology - Revision history

Vipul at 16:57, 20 December 2010

2010-12-20T16:57:58Z

← Older revision		Revision as of 16:57, 20 December 2010
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	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.		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.

			==Relation with other function space topologies==

			{\| class="sortable" border="1"
			! Topology !! Meaning !! Relationship with compact-open topology
			\|-
			\| [[topology of pointwise convergence]] \|\| topology chosen such that a sequence of functions converges iff it converges pointwise; equivalently, the [[subspace topology]] inherited from the [[product topology]] on the space <math>Y^X</math> of all functions. \|\| ?
			\|-
			\| [[topology of uniform convergence]] \|\| \|\|
			\|-
			\| [[topology of compact convergence]] \|\| \|\|
			\|}

			==Particular cases==

			{\| class="sortable" border="1"
			! Case for <math>X</math> !! Case for <math>Y</math> !! Overall conclusion !! Necessarily equals topology of pointwise convergence? !! Necessarily equals topology of uniform convergence? (if that makes sense)
			\|-
			\| [[one-point space]] \|\| anything \|\| canonically identified with <math>Y</math>, identification is a homeomorphism \|\| Yes \|\| Yes
			\|-
			\| [[discrete space]] \|\| anything \|\| canonically identified with <math>Y^X</math>, space of all functions from <math>X</math> to <math>Y</math>, with the [[product topology]] \|\| Yes \|\| Yes (if that makes sense)
			\|-
			\| anything \|\| [[T0 space]] \|\| also a [[T0 space]]: see [[function space to T0 space is T0 under compact-open topology]] \|\| No \|\| No
			\|-
			\| anything \|\| [[T1 space]] \|\| also a [[T1 space]]: see [[function space to T1 space is T1 under compact-open topology]] \|\| No \|\| No
			\|-
			\| anything \|\| [[Hausdorff space]] \|\| also a [[Hausdorff space]]: see [[function space to Hausdorff space is Hausdorff under compact-open topology]] \|\| \|\|
			\|-
			\| anything \|\| [[regular space]] \|\| also a [[regular space]]: see [[function space to regular space is regular under compact-open topology]] \|\| \|\|
			\|-
			\| anything \|\| [[completely regular space]] \|\| also a [[completely regular space]]: see [[function space to completely regular space is completely regular under compact-open topology]] \|\| \|\|
			\|}

Vipul at 16:18, 20 December 2010

2010-12-20T16:18:07Z

← Older revision		Revision as of 16:18, 20 December 2010
Line 5:		Line 5:
	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.

Vipul: 1 revision

2008-05-11T19:41:05Z

1 revision

← Older revision	Revision as of 19:41, 11 May 2008
(No difference)

Vipul: New page: {{function space topology}} ==Definition== Suppose $X$ and $Y$ are topological spaces. The '''compact-open topology''' is a topology we can define on the space ...

2008-02-02T01:02:48Z

New page: {{function space topology}} ==Definition== Suppose <math>X</math> and <math>Y</math> are topological spaces. The '''compact-open topology''' is a topology we can define on the space ...

New page

{{function space topology}}

==Definition==

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.

Compact-open topology - Revision history

Vipul at 16:57, 20 December 2010

Vipul at 16:18, 20 December 2010

Vipul: 1 revision

Vipul: New page: {{function space topology}} ==Definition== Suppose X and Y are topological spaces. The '''compact-open topology''' is a topology we can define on the space ...

Vipul: New page: {{function space topology}} ==Definition== Suppose $X$ and $Y$ are topological spaces. The '''compact-open topology''' is a topology we can define on the space ...