Open sets appear directly in the definition of a topological space.
It next seems that closed sets are needed. Suppose is a
topological space. A subset
is defined to be a
*closed set* if and only if
is an open set. Thus,
the complement of any open set is closed, and the complement of any
closed set is open. Any closed interval, such as , is a closed
set because its complement,
, is an open
set. For another example, is an open set; therefore,
is a closed set. The
use of ``'' may seem wrong in the last expression, but ``''
cannot be used because and do not belong to
.
Thus, the use of ``'' is just a notational quirk.

Are all subsets of either closed or open? Although it appears
that open sets and closed sets are opposites in some sense, the answer
is *no*. For
, the interval is neither open
nor closed (consider its complement:
is
closed, and
is open). Note that for any topological
space, and are both open and closed!

Steven M LaValle 2012-04-20