Metric is jointly continuous - Revision history

Vipul at 21:25, 19 July 2008

2008-07-19T21:25:09Z

← Older revision		Revision as of 21:25, 19 July 2008
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	==Statement==		==Statement==

	Let <math>(X,d)</math> be a [[metric space]]. Then <math>X</math> is also a topological space in the induced topology, and we can consider the metric as a map of topological spaces <math>d:X \times X \to \R</math>. This map is jointly continuous, i.e. it is continuous from <math>X \times X</math> given the product topology.		Let <math>(X,d)</math> be a [[metric space]]. Then <math>X</math> is also a [[topological space]] in the induced topology, and we can consider the metric as a map of topological spaces <math>d:X \times X \to \R</math>. This map is jointly continuous, i.e. it is continuous from <math>X \times X</math> given the product topology.

			==Definitions used==

			===Metric space===

			===Topology induced by a metric===

			===Product topology===

			===Continuous map===

	==Proof==		==Proof==

	It suffices to show that inverse images of open subsets of the form <math>(-\infty,a)</math> and <math>(b,\infty)</math> are open subsets of <math>X \times X</math>. We will use the triangle inequality to prove this.		It suffices to show that inverse images of open subsets of the form <math>(-\infty,a)</math> and <math>(b,\infty)</math> are open subsets of <math>X \times X</math>. We will use the triangle inequality to prove this.

	~~{{fillin}}~~

Vipul: 2 revisions

2008-05-11T19:53:51Z

2 revisions

← Older revision	Revision as of 19:53, 11 May 2008
(No difference)

Vipul at 02:23, 24 January 2008

2008-01-24T02:23:13Z

← Older revision		Revision as of 02:23, 24 January 2008
Line 6:		Line 6:

	It suffices to show that inverse images of open subsets of the form <math>(-\infty,a)</math> and <math>(b,\infty)</math> are open subsets of <math>X \times X</math>. We will use the triangle inequality to prove this.		It suffices to show that inverse images of open subsets of the form <math>(-\infty,a)</math> and <math>(b,\infty)</math> are open subsets of <math>X \times X</math>. We will use the triangle inequality to prove this.

			{{fillin}}

Vipul: New page: ==Statement== Let $(X,d)$ be a metric space. Then $X$ is also a topological space in the induced topology, and we can consider the metric as a map of topological...

2008-01-24T02:23:01Z

New page: ==Statement== Let <math>(X,d)</math> be a metric space. Then <math>X</math> is also a topological space in the induced topology, and we can consider the metric as a map of topological...

New page

==Statement==

Let <math>(X,d)</math> be a [[metric space]]. Then <math>X</math> is also a topological space in the induced topology, and we can consider the metric as a map of topological spaces <math>d:X \times X \to \R</math>. This map is jointly continuous, i.e. it is continuous from <math>X \times X</math> given the product topology.

==Proof==

It suffices to show that inverse images of open subsets of the form <math>(-\infty,a)</math> and <math>(b,\infty)</math> are open subsets of <math>X \times X</math>. We will use the triangle inequality to prove this.

Metric is jointly continuous - Revision history

Vipul at 21:25, 19 July 2008

Vipul: 2 revisions

Vipul at 02:23, 24 January 2008

Vipul: New page: ==Statement== Let (X,d) be a metric space. Then X is also a topological space in the induced topology, and we can consider the metric as a map of topological...

Vipul: New page: ==Statement== Let $(X,d)$ be a metric space. Then $X$ is also a topological space in the induced topology, and we can consider the metric as a map of topological...