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In the usual topology on the integers, i.e. the Euclidean metric topology, should $\{1,2,3,4\} \in \mathbb{Z}$ be clopen? If the usual topology is measured with real numbers, even regarding the integers, then the subset both contains it's limit points and should include an open subset around each point without including an element out of the subset. However I wasn't sure if the usual topology should be measured with real numbers regarding the integers.

If it should be measured with integers then the set wouldn't be closed which is not really my intuition but I still have to ask.

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So really the answers to your questions are 'yes' and 'yes'. You would use the real numbers to measure the distance between points under the Euclidean metric. And the integers, as a subspace of the reals, with the usual metric, have the discrete topology and so every set is open and closed.

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