October 3rd, 2011, 08:48 AM  #1 
Newbie Joined: Aug 2011 Posts: 9 Thanks: 0  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? 
October 3rd, 2011, 10:17 AM  #2 
Senior Member Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0  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?

October 3rd, 2011, 11:23 AM  #3 
Member Joined: Jun 2011 From: California Posts: 82 Thanks: 3 Math Focus: Topology  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.

October 9th, 2011, 06:11 AM  #4  
Senior Member Joined: Apr 2010 Posts: 451 Thanks: 1  Re: Open Subset Quote:
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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