1. ## Open set

Hi! I have problems with this exercises

Let $(\mathbb{R^2},d)$ where d is Euclidean metric. Let $A= \{ (x,y) | -1<x<1 , -1<y<1\}$. Show that $A$ is an open set

Thanks

2. ## Re: Open set

Originally Posted by cristianoceli
Let $(\mathbb{R^2},d)$ where d is Euclidean metric. Let $A= \{ (x,y) | -1<x<1 , -1<y<1\}$.
Show that $A$ is an open set.
To show that a set is open you must show that $\forall (x,y)\in A$ then $\exists\mathscr{B}_r(x,y)\subset A$
So if $(x,y)\in A$ by definition $-1<x<1~\&~-1<y<1$
Now suppose that $\delta=\frac{1}{2}\min\{x+1,1-x,y+1,1-y\}$
you must show that $\mathscr{B}_{\delta}(x,y)\subset A$

3. ## Re: Open set

Originally Posted by Plato
To show that a set is open you must show that $\forall (x,y)\in A$ then $\exists\mathscr{B}_r(x,y)\subset A$
So if $(x,y)\in A$ by definition $-1<x<1~\&~-1<y<1$
Now suppose that $\delta=\frac{1}{2}\min\{x+1,1-x,y+1,1-y\}$
you must show that $\mathscr{B}_{\delta}(x,y)\subset A$
in case it's not obvious, (and it's not)

$\mathscr{B}_r(x,y) = \{(u,v) \ni d[(x,y),(u,v)]< r\}$

i.e. a open disk centered at $(x,y)$ of radius $r$

4. ## Re: Open set

Originally Posted by Plato
To show that a set is open you must show that $\forall (x,y)\in A$ then $\exists\mathscr{B}_r(x,y)\subset A$
So if $(x,y)\in A$ by definition $-1<x<1~\&~-1<y<1$
Now suppose that $\delta=\frac{1}{2}\min\{x+1,1-x,y+1,1-y\}$
you must show that $\mathscr{B}_{\delta}(x,y)\subset A$
Why do you choose $\delta=\frac{1}{2}\min\{x+1,1-x,y+1,1-y\}$ ?

5. ## Re: Open set

x+1 is the distance from the point (x, y) to the boundary x= -1. 1- x is the distance from the point (x, y) to the line x= 1, 1- y is the distance from the point (x, y) to the line y= 1, and 1- y is the distance from the point (x, y) to the line y= -1. Take the smallest of those and divide by two to get a radius smaller than any of those distances so the circle around (x, y) is inside the set. Of course, you could multiply by any number less than 1, not just 1/2.

Ok thanks!!!