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October 3rd, 2011, 09:48 AM   #1
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Open Subset

Is there any subset of R that is an open subset of R^2?

My answer is yes. Example (0,1) member of R. It open square (0,1) * (0,1) is an open subset of R^2 right?
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October 3rd, 2011, 11:17 AM   #2
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Re: Open Subset

This question is badly phrased. An element of R cannot be thought of as an element of R^2 unless some embedding is specified. If it means the embedding then what happens if you consider an open ball around any point in the range of the embedding?
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October 3rd, 2011, 12:23 PM   #3
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Re: Open Subset

And yes, the square is open in equipped with the Euclidean topology. is open in equipped with the Euclidean topology, so the Cartesian product of with itself is open in with the Euclidean topology.
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October 9th, 2011, 07:11 AM   #4
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Re: Open Subset

Quote:
Originally Posted by fienefie
Is there any subset of R that is an open subset of R^2?

My answer is yes. Example (0,1) member of R. It open square (0,1) * (0,1) is an open subset of R^2 right?

The only subset of R which is an open set in IS THE EMPTY SET.

No other subset of R is an open set in .

To justify the above let us take the definition of the open set:

A set S is open in E iff ,for every point x ,of S there exists an open ball ( with center x and radius r) which lies entirely within S.

Now an open ball with center x, and radius r in , denoted by B(x,r) = { a: d(x,a)<r} .d is any metric in

Now the open balls in ARE usaly circles or squares.

According to the above definitions no subset of R is an open set in .

Let us take your example.Can we,for every point ,x ,of (0,1), find a circle or square that will lie entirely within (0,1)??

The empty set how ever is an open set in
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