I would like a little clarification in how to prove that the k topology on R is strictly finer than the standard topology on R. They have a proof of this in Munkres' book. I know how to prove that its finer, but the part that shows it to be strictly finer im not sure. It says given the basis element B = (-1,1) - K for T'' (the k topology), there is no open interval that contains 0 and lies in B. If what it says is what i think then i can think of many counterexamples, for example: use the element 1/2 of K. (-1,1)-K = (-3/2,1/2). Use the open interval (-1/3, 1/3) which lies in B right?