# how to prove a closed set in Rⁿ

• May 23rd 2011, 11:15 PM
sorv1986
how to prove a closed set in Rⁿ
hi!

what is the easiest way to prove that the set A is closed and the set B is not closed?

1). A={(x,y) ∈ R²: x²+y²=1}
and
2). B={(x,y) : ∈ R²: x² - y²=1}

• May 23rd 2011, 11:18 PM
Drexel28
Quote:

Originally Posted by sorv1986
hi!

what is the easiest way to prove that the set A is closed and the set B is not closed?

1). A={(x,y) ∈ R²: x²+y²=1}
and
2). B={(x,y) : ∈ R²: x² - y²=1}

I mean, the easiest way would be to note that $f:\mathbb{R}^2\to\mathbb{R}$, $f(x,y)=x^2+y^2$ is continuous (it's just a polynomial) and $A=f^{-1}({1})$ and for the second one are you sure it isn't closed?