## K topology strictly finer than standard topology

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?
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 Recognitions: Homework Help Science Advisor No, all the elements 1, 1/2, 1/3, 1/4, 1/5, ... are outside of B. Any open interval around 0 has to have one of these fractions. Think about it. An open interval around 0 must be of the form (a, b) with a < 0 < b. If b > 1, then choose n = 2. Clearly, 1/2 is in (a, b) but it's not in B. If b < 1, take n = floor(1/b). Then 1/n is in (a,b) but not in B.
 ok i think i know what my problem was. I took (-1,1) - K to mean the set of all x-1/n between -1 and 1, where n is a positive integer and x is real. I guess the minus K actually means exclude any 1/n for any positive integer n from the interval (-1,1). Yes, in that case, any open interval in there would have to contain some 1/n 's, and therefore is not in B.

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