- a set $A$ which intersects every nonempty open subset of $X$.
If $U\subset X$, a set $A\subset X$ is called dense in $U$ if $A\cap U$ is a dense set in the subspace topology of $U$. When $U$ is open this is equivalent to the requirement that the closure (in $X$) of $A$ contains $U$.
A set which is not dense in any non-empty open subset of a topological space $X$ is called nowhere dense.
A set which consists of limit points is called dense-in-itself.
Everywhere-dense set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Everywhere-dense_set&oldid=28111