Question

Show,The set U = f(x; y) 2 R2 j x2 + y2 <=1, and x > 0g is open in B(0; 1)
where the norm on R2 is the Euclidean norm.

Answer 1

As, the question is not clearly defined, I am assuming the question goes like this otherwise it doesn't make sense.

$$U=\{(x,y)\in \mathbb{R}^2:x^2+y^2\leq 1\&x>0\}$$

We have to show $U$ is open in $\overline{B}(0,1)$ where $B(0,1)$ is open ball centered at with radius $1$ and of course norm is Euclidean norm.

Clearly, our induced matrix space is $\overline{B}(0,1)$ where metric is induced from $(\mathbb{R}^2,|| \: ||_2)$

Let, for any

$$v_n=(x_n,y_n)\in U$$

consider $r_n=1-\frac{1}{n}>0$ ,Thus,consider the open ball $B'(v_n,r_n)$ ,Hence

$$U=\cup_{n=1}^{\infty}B'(v_n,r_n)$$

Thus, we are done.