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Are there sets in the K-Topology that aren’t open in the standard topology?

Of course this sounds like some materials from Munkre's book, but I still do not get why K-topology (collection of sets of the form $\{(a,b) \setminus K=\{\frac{1}{n}| n \in \mathbb{N} \}\}$ is finer than the standard topology.

It is easy to show that the lower limit topology is finer than the standard topology, but I am stuck on the case for K-topology.

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Daniel
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  • I just wonder is there a way to prove that K-topology is strictly finer than standard topology in a direct way, i.e not using the example in the thread given? – Daniel Jan 31 '13 at 00:06
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    Exhibiting a set that is open in the $K$-topology but not in the Euclidean topology is about as direct as a proof can get! – Brian M. Scott Jan 31 '13 at 02:51

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