e exists in (-1,1) and =.9999999999...
f exists in (-2,2) and =1.999999999...
in this case, if g=min{e,f} then it works
i get that, but what i don't get is when one is not a subset of the other and they just overlap a little (e.g. (-3,1) and (0,4)
i have e>0 and f>0, and i tried g=min{e,f} but that doesn't work because it doesn't guarantee that g will be small enough to only exist in the intersection of X and Y
it's the number line with shaded between -1 and 1 with circles at the end to indicate that the end points are not included in the set
i'm thinking more of a Venn Diagram for a picture.
i'm trying to use the definition of intersection in my proof, but i can't seem to get it right. i can prove the x is an element in X or Y, but it's the and that's tripping me up.
i am having a major brain fart. ok, so if I'm only looking at one point x that is an element of the intersection of X and Y, and e>0.
then the i have to consider the cases:
x is an element in X or x is an element in Y and they do not over lap, then the intersection is obviously open because...
i have been able to prove that there is an e such that (x-e,x+e) is wholly in X and there is an f such that (y-f,y+f) is wholly in Y, I'm having a problem proving that there exists a g such that (z-g,z+g) exists only in the intersection of X and Y. i can't figure out what g should equal to make...
the open sets are in R, but i need to prove that the intersection of just two open sets is open. once i have that, proving the intersection of a finite number of open sets is easy. I'm trying to use an open ball in the proof.
i'm at a loss.