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 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 $x\compose[10]{\to}{\tiny\vdash}(x,0),$ 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 $(0, 1) \times (0, 1)$ is open in $\mathbb{R}^2$ equipped with the Euclidean topology. $(0, 1)$ is open in $\mathbb{R}$ equipped with the Euclidean topology, so the Cartesian product of $(0, 1)$ with itself is open in $\mathbb{R}^2$ with the Euclidean topology.
October 9th, 2011, 06:11 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 $R^2$ IS THE EMPTY SET.

No other subset of R is an open set in $R^2$.

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 $R^2$, denoted by B(x,r) = { a: d(x,a)<r} .d is any metric in $R^2$

Now the open balls in $R^2$ ARE usaly circles or squares.

According to the above definitions no subset of R is an open set in $R^2$.

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 $R^2$

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