Is the Square {(x,y) : |x|< 1, |y|< 1} an Open Set in the Metric Space (R^2, d)?

[Pyroadept]

Homework Statement

In the metric space (R^2, d), where d is the standard euclidean metric, show that the square {(x,y) : |x|< 1, |y|< 1} is open.

Homework Equations

A set is closed if and only if its complement is open.

A set is open if every point in the set is an interior point, that is, you can construct an open ball around the point such that the open ball is contained in the set.

The Attempt at a Solution

Hi everyone,

I'm really stuck. Intuitively I can of course see it's right and if I draw a picture it looks right. My problem is with constructing the rigid proof; I don't have a clue where to begin. If I draw a unit circle within the square I could show that it's an open set as I know how to show an open ball is an open set. But does the proof for the square work along a similar principle? But how do I apply it?
I know you guys can't give me the full solution, but if you could even point me in the right direction I would really appreciate it!

Thanks in advance for any help. :)

 

[tiny-tim]

Hi Pyroadept!

Just take a typical point (x,y), and find a ball round it small enough to fit into the square.