Compact-open topology: Difference between revisions

Latest revision as of 16:57, 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.

Relation with other function space topologies

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 $Y^{X}$ of all functions.	?
topology of uniform convergence
topology of compact convergence

Particular cases

Case for $X$	Case for $Y$	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 $Y$ , identification is a homeomorphism	Yes	Yes
discrete space	anything	canonically identified with $Y^{X}$ , space of all functions from $X$ to $Y$ , 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

@@ Line 6: / Line 6: @@
 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]] || ||
+|}