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- February 19th 2010, 11:33 AM #1

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## What is the Interoir of the set QxQ?

Given the metric space (M,d) where M =

**R**x**R**and d(x,y) = the Euclidean metric, what is the interior of the subset of M,**Q**x**Q**?

I realize it is the empty set because there are points that lie in the open ball of any element of**Q**x**Q**that are not actually in the set**Q**x**Q**.

What I am struggling with is giving a formal, reasonable justification of why the interior is the empty set.

Any help or suggestions to get me thinking in a good direction would be greatly appreciated. Thank you so much, in advance!

- February 19th 2010, 11:45 AM #2

- February 19th 2010, 11:50 AM #3

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- February 19th 2010, 02:04 PM #4
To prove what

**Plato**has said, try to prove that if are dense subsets of the topological spaces then is dense in under the product topology. Now since is dense in we know that is dense in . Similarly, is dense in and so given any for any neighborhood of there exists points of both and since the latter is a subset of it follows that . Since was arbitrary it follows that . And, as**Plato**pointed out we have that no interior point can be a boundary point, and so

Note: The product topology coincides with the usual topology on