Let 
be a subset of a metric
space. Then the set 
is open if every point in 
has a neighborhood lying
in the set. An open set of radius 
and center 
is the set of all points 
such that 
, and is denoted 
. In one-space, the open set is an open
interval. In two-space, the open set is a disk. In three-space,
the open set is a ball.
More generally, given a topology (consisting of a set 
and a collection of subsets 
), a set is said to be open if it is
in 
. Therefore, while it is not possible
for a set to be both finite and open in the topology
of the real line (a single point is a closed
set), it is possible for a more general topological set
to be both finite and open.
The complement of an open set is a closed set. It is possible for a set to be neither open nor closed, e.g.,
the half-closed interval ![(0,1]](/images/equations/OpenSet/Inline12.svg)
.
