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In: Advanced Math

Let q and p be natural numbers, and show that the metric on Rq+p is equivalent...

Let q and p be natural numbers, and show that the metric on R^q+p is equivalent to the metric it has if we identify R^q+p with R^q × R^p.

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Expert Solution

We want to show, the metric topology on is equivalant to product topology of . We can identify points of and in natural way. Now pick . Take an open ball arround of radius . Now it contains points , such that . That is same as saying, Now, and . Take balls and of radius arround and . And consider . Now, belong to the product topology of .

contains all points such that and . Then in , . So, .

Next we need to show for , is a ball centered at of radius , and is a ball centered at of radius , we can find a real number , such that . Now, for , and for ,. With out loss of generality, let be minimum of . Then take . Then . That imply and . So, for this , .

So these two topologies are equivalant.

Colby Messinger answered 1 year ago

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