A set
in a first-countable space is dense in
if
, where
is the set of limit points of
. For example, the rational numbers are dense in the reals.
In general, a subset
of
is dense if its set closure
.
A real number
is said to be
-dense
iff, in the base-
expansion of
, every possible finite string of consecutive digits appears.
If
is
-normal, then
is also
-dense. If, for some
,
is
-dense, then
is irrational. Finally,
is
-dense
iff the sequence
is dense (Bailey and Crandall 2001, 2003).