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Let R be the set of all real numbers and let K={1/n, n is a natural number}. Generate a topology on R by taking as basis all open intervals (a,b) and all sets of the form (a,b)-K (the set of all elements in (a,b) that are not in K). The topology generated is known as the K-topology on R.

K-topology satisfies the Hausdorff axiom. I don't know how to prove it at all. This is my HW. Please help me.

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    @Jihyun Kim: You should edit that into your question, so it's more visible.2011-10-23

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Hint: We want to show that if $a$ and $b$ are distinct points of $\mathbb{R}$, they can be "separated" by disjoint sets that are open in the $K$-topology. Now use the fact that every set which is open in the ordinary topology on $\mathbb{R}$ is open in the $K$-topology.