# What is the name of the complement of a nowhere dense set?

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?

• Is “everywhere dense” what you’re looking for? – MPW Sep 26 '18 at 12:15
• @MPW I think everywhere dense is equivalent to "dense", which is weaker than the notion of "a set with dense interior" – High GPA Sep 26 '18 at 12:18
• I've heard "co-nowhere dense" in a talk, but I don't think that's common terminology. – Noah Schweber Sep 26 '18 at 15:46
• 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". – Dave L. Renfro Sep 27 '18 at 19:57
• 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. – Dave L. Renfro Sep 30 '18 at 19:33