Let A,B be subsets of a normed linear space X.
Show that 1) If A or B is open then A+B is open.
2) If A and B both are compact then A+B is compact..
Any help appreciated..
1)are you sure it's not if A and B is open?
2) Assuming both A and B are both compact, then for any open cover covering A we can find a finite subset G of S s.t is also an open cover for A. same goes for B... Is the union of two finite sets finite?
Hey for the Part 2) if I define a map f:A*B-->A+B by f(a,b) = a+b then f is a continous map..Also A*B is compact..all i need to show is f is surjective..then i can say that A+B is also compact..
Please help me in the surjection part!
Hey for the Part 2) if I define a map f:A*B-->A+B by f(a,b) = a+b then f is a continous map..Also A*B is compact..all i need to show is f is surjective..then i can say that A+B is also compact..
Please help me in the surjection part!
Yes, that's a neat way to do part 2), and the surjectivity is really obvious! In fact, by definition any element of A+B is of the form a+b, for some a in A and b in B. So it is equal to f(a,b). That's all there is to it.