1.Prove B is not open set. (or find counterexample.) 2.What is int(B)? Solution

Question

1.Prove B is not open set. (or find counterexample.)

2.What is int(B)?

Explanation / Answer

So B = set of fns which go from R to positive R. You have asked to prove that B is not open. So to do this we need a func. g in B st any neighborhood of g cannot lie in B Consider this function : g(x) = e^(-x) now g(x) > 0 for all x consider any epsilon > 0 we will prove there exists f st |g-f| infty f(x) = -epsilon/2 so f is not in B hence any nbd of g cannot lie in B Hence B is not open. (I dont think B is closed either) b) int B = set of all interior points it would be this : int B = {set of all fns which are away from zero by a positive amount} = {f| f(x) > c for some c>0} = {f | inf(f(x) over x in R) >0} message me if you have any doubts.

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