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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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No. One would just say that the set is neither open nor closed. – Adam Smith Oct 18 '11 at 18:47
I would call it set. – Mariano Suárez-Alvarez Oct 18 '11 at 18:50
@Mariano : But a set could (possibly) be open or closed, while MikeC wants it to be neither :). – Adam Smith Oct 18 '11 at 18:53
I don't know of such a name. We name things when they are referenced frequently enough, but sets which are neither closed nor open are not of much interested in topology, so there is no need to name them. Essentially, topology interests itself with the open subsets of the space, and their complements, the closed sets. The sets that are not closed or opened are, by the nature of topology, "not interesting." – Thomas Andrews Oct 18 '11 at 18:53
Maybe it is time to agree to call such sets ajar. – Will Oct 18 '11 at 19:18
up vote 4 down vote accepted

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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Topological terminology is used outside of topology proper. In my field of research convex cones that are neither open nor closed are 'interesting' and 'natural', whereas clopen sets arise only as edge cases. So while your answer may be true, the last sentence should perhaps be qualified with 'in topology'. – equaeghe Mar 2 '12 at 10:32
@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. – Rasmus Mar 2 '12 at 12:55

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