# Dense set

(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