(adsbygoogle = window.adsbygoogle || []).push({}); Is this set really not closed ?!!?!

Ok, consider the metric space [itex]\mathbb{R}^2[/itex] armed with the pythagorean metric [itex]d(x,y) = ||x-y||[/itex] and B, the closed ball of radius 1 centered on the origin: [itex]B = B_1(0) \cup \partial B_1(0)[/itex]. Now construct the metric space composed of this closed ball and the appropriate restriction on the pythagorean metric.

Now consider D, a closed ball centered on the origin but of radius r<1, and ask the question: is D closed in B? D is closed iff [itex]D^c = B \slash D[/itex] is open in B. But it is not, for consider a point on the border of B ([itex]x_0 \in \partial B_1(0)[/itex]). No ball with x_0 as center is contained entirely in B. Hence D is not closed in B.

Is that correct?!

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# Homework Help: Is this set really not closed ? ?

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