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In topology, a set which is both open and closed is often called "clopen." Is there a commonly used term for sets which are neither open nor closed?

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    Maybe it is time to agree to call such sets ajar.2011-10-18

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No there isn't. The reason that there is an extra name for clopen sets is that they are very useful and natural to consider, for example in the context of questions concerning connectivity. This does not seem to be the case for non-open non-closed sets.

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    @equaeghe: I am not saying that sets which are neither open nor closed are necessarily uninteresting. I am saying that the property of being neither open nor closed is not an interesting property *per se*.2012-03-02