- #1
stunner5000pt
- 1,461
- 2
I am not well 'accustomed' to these kind of proofs so please bear with my stupidity
Suppose U and V are both open subsets in Rn. Prove that U intersection V and U union V are open as well.
Dfeinition of open is that you cna center a ball about a point a in a set such that that ball is completely contained in the set.
So let there be a ball with center a radius delta in U such that
[tex] B(a,\delta_{1}) = { x: ||x-a||< \delta_{1}} [/tex] for U and
[tex] B(b,\delta_{2}) = { y: ||y-b||< \delta_{2}} [/tex] for V
now the for the union
[tex] B(c, \delta) = {z: ||z-c|| < \delta} [/tex]
and c belongs to the union of U and V. and picking delta = min (delta 1, delta 2) we can say that the ball is completely contained in U union V and the union is open?
If this correct we can move to the intersection...
do i do a similar procedure? Please help?
Thank you for you help and advice!
Suppose U and V are both open subsets in Rn. Prove that U intersection V and U union V are open as well.
Dfeinition of open is that you cna center a ball about a point a in a set such that that ball is completely contained in the set.
So let there be a ball with center a radius delta in U such that
[tex] B(a,\delta_{1}) = { x: ||x-a||< \delta_{1}} [/tex] for U and
[tex] B(b,\delta_{2}) = { y: ||y-b||< \delta_{2}} [/tex] for V
now the for the union
[tex] B(c, \delta) = {z: ||z-c|| < \delta} [/tex]
and c belongs to the union of U and V. and picking delta = min (delta 1, delta 2) we can say that the ball is completely contained in U union V and the union is open?
If this correct we can move to the intersection...
do i do a similar procedure? Please help?
Thank you for you help and advice!