Metric is jointly continuous: Difference between revisions

Revision as of 02:23, 24 January 2008

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 spaces $d : X \times X \to R$ . This map is jointly continuous, i.e. it is continuous from $X \times X$ given the product topology.

Proof

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

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Revision as of 02:23, 24 January 2008 (view source) Vipul (talk \| contribs) (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...)		Revision as of 02:23, 24 January 2008 (view source) Vipul (talk \| contribs) No edit summary Newer edit →
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	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.

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