Dense set

2020 Mathematics Subject Classification: Primary: 54A05 [MSN][ZBL]

A subset $A$ of a topological space $X$ is dense for which the closure is the entire space $X$ (some authors use the terminology everywhere dense). A common alternative definition is:

• 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.