# Dense set

From Encyclopedia of Mathematics

(Redirected from Everywhere-dense set)

2010 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.

**How to Cite This Entry:**

Everywhere-dense set.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Everywhere-dense_set&oldid=28111

This article was adapted from an original article by A.A. Mal'tsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article