Vipul at 06:31, 15 May 2015

2015-05-15T06:31:52Z

← Older revision		Revision as of 06:31, 15 May 2015
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	==Relation with separate continuity==		==Relation with separate continuity==

	''Jointly continuous'' is typically contrasted with the notion of a [[separately continuous map]]. A map <math>f:A_1 \times A_2 \times \dots \times A_n \to C</math> is ''separately'' continuous if it is continuous from the [[slice topology]] on <math>A_1 \times A_2 \times \dots A_n</math>. More explicitly, it is continuous in each coordinate fixing the values of all the other coordinates. The slice topology is a [[finer topology]] (often, but not necessarily, strictly finer) than the ~~slice~~ topology. Thus, joint continuity is a stronger (and in some cases, strictly stronger) condition than separate continuity.		''Jointly continuous'' is typically contrasted with the notion of a [[separately continuous map]]. A map <math>f:A_1 \times A_2 \times \dots \times A_n \to C</math> is ''separately'' continuous if it is continuous from the [[slice topology]] on <math>A_1 \times A_2 \times \dots A_n</math>. More explicitly, it is continuous in each coordinate fixing the values of all the other coordinates. The slice topology is a [[finer topology]] (often, but not necessarily, strictly finer) than the product topology. Thus, joint continuity is a stronger (and in some cases, strictly stronger) condition than separate continuity.

	Joint continuity is the correct condition in most circumstances. For instance, if <math>f:A \times A \to C</math> is separately continuous, we cannot be sure whether <math>f \circ \delta</math> is continuous, where <math>\delta:A \to A \times A</math> is the diagonal embedding. More generally, for a separately continuous map, we cannot guarantee continuity under a ''simultaneous'' change of both coordinates.		Joint continuity is the correct condition in most circumstances. For instance, if <math>f:A \times A \to C</math> is separately continuous, we cannot be sure whether <math>f \circ \delta</math> is continuous, where <math>\delta:A \to A \times A</math> is the diagonal embedding. More generally, for a separately continuous map, we cannot guarantee continuity under a ''simultaneous'' change of both coordinates.

	Thus, all definitions that involve continuity from products use joint continuity. Examples include the definition of [[topological magma]], [[topological monoid]], and [[topological group]], the definition of [[homotopy]] between maps, and all the definitions/concepts arising from homotopy.		Thus, all definitions that involve continuity from products use joint continuity. Examples include the definition of [[topological magma]], [[topological monoid]], and [[topological group]], the definition of [[homotopy]] between maps, and all the definitions/concepts arising from homotopy.

Vipul: Created page with '==Definition== Suppose $A_1, A_2, \dots, A_n, C$ are topological spaces. A set map $f:A_1 \times A_2 \times \dots \times A_n \to C$ is termed a '''join...'

2010-12-20T22:16:04Z

Created page with '==Definition== Suppose <math>A_1, A_2, \dots, A_n, C</math> are topological spaces. A set map <math>f:A_1 \times A_2 \times \dots \times A_n \to C</math> is termed a '''join...'

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==Definition==

Suppose <math>A_1, A_2, \dots, A_n, C</math> are [[topological space]]s. A set map <math>f:A_1 \times A_2 \times \dots \times A_n \to C</math> is termed a '''jointly continuous map''' if it satisfies the following equivalent conditions:

# <math>f</math> is a [[defining ingredient::continuous map]] from <math>A_1 \times A_2 \times \dots \times A_n</math> equipped with the [[defining ingredient::product topology]].
# Suppose <math>X</math> is a topological space and <math>g_i:X \to A_i</math> are all [[continuous map]]s. Then, the map <math>h:X \to C</math> given by <math>h(x) = f(g_1(x),g_2(x),\dots,g_n(x))</math> is a continuous map. (This should be true for ''every'' choice of <math>X</math> and <math>g_i</math>s).

==Relation with separate continuity==

''Jointly continuous'' is typically contrasted with the notion of a [[separately continuous map]]. A map <math>f:A_1 \times A_2 \times \dots \times A_n \to C</math> is ''separately'' continuous if it is continuous from the [[slice topology]] on <math>A_1 \times A_2 \times \dots A_n</math>. More explicitly, it is continuous in each coordinate fixing the values of all the other coordinates. The slice topology is a [[finer topology]] (often, but not necessarily, strictly finer) than the slice topology. Thus, joint continuity is a stronger (and in some cases, strictly stronger) condition than separate continuity.

Joint continuity is the correct condition in most circumstances. For instance, if <math>f:A \times A \to C</math> is separately continuous, we cannot be sure whether <math>f \circ \delta</math> is continuous, where <math>\delta:A \to A \times A</math> is the diagonal embedding. More generally, for a separately continuous map, we cannot guarantee continuity under a ''simultaneous'' change of both coordinates.

Thus, all definitions that involve continuity from products use joint continuity. Examples include the definition of [[topological magma]], [[topological monoid]], and [[topological group]], the definition of [[homotopy]] between maps, and all the definitions/concepts arising from homotopy.

Jointly continuous map - Revision history

Vipul at 06:31, 15 May 2015

Vipul: Created page with '==Definition== Suppose A_1, A_2, \dots, A_n, C are topological spaces. A set map f:A_1 \times A_2 \times \dots \times A_n \to C is termed a '''join...'

Vipul: Created page with '==Definition== Suppose $A_1, A_2, \dots, A_n, C$ are topological spaces. A set map $f:A_1 \times A_2 \times \dots \times A_n \to C$ is termed a '''join...'