Let $A$ be a nowhere dense set, what is the name of $A^c$? It seems much stronger than the definition of dense set. I think if we assume $A$ is a close set, then $A$ is nowhere dense iff $A^c$ is dense. But what about the general case?

I found a long name for this: a dense set with dense interior. Does there exist a fancier name, probably named after a person?

  • $\begingroup$ Is “everywhere dense” what you’re looking for? $\endgroup$ – MPW Sep 26 '18 at 12:15
  • $\begingroup$ @MPW I think everywhere dense is equivalent to "dense", which is weaker than the notion of "a set with dense interior" $\endgroup$ – High GPA Sep 26 '18 at 12:18
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    $\begingroup$ I've heard "co-nowhere dense" in a talk, but I don't think that's common terminology. $\endgroup$ – Noah Schweber Sep 26 '18 at 15:46
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    $\begingroup$ Incidentally, there are stronger notions of density than what "complement of nowhere dense" gives you, namely notions that involve being the complement of a porous set (there are many different notions of this), and some authors use the term "plump set" for the corresponding open sets that these porous sets are the complements of. For example, see the google search "plump" + "uniform domain". $\endgroup$ – Dave L. Renfro Sep 27 '18 at 19:57
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    $\begingroup$ Regarding my first comment above, just two days later I happened to see a Stack Exchange question in which "generic set" is used to mean an arbitrarily chosen set. $\endgroup$ – Dave L. Renfro Sep 30 '18 at 19:33

A nowhere dense set is one whose closure has an empty interior.

Therefore the complement of a nowhere dense set is one whose interior has a closure that is the entire space.

Or, in other words, a set whose interior is dense.

Or, in yet other words, a superset of a dense open set.


Generic set is sort of usual. See the topology section of the linked page. But in Baire spaces this name is also used for co-meagre sets. I personally like generic set.

  • $\begingroup$ I agree. However it is not precise because the superset of a dense open set is stronger than "generic" which is usually just dense open set. $\endgroup$ – High GPA Sep 28 '18 at 8:33

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