boundary of an open set is nowhere dense
This entry provides another example of a nowhere dense set.
Let . Since , it is closed, so all we need to show is that has empty interior . First notice that , since is open. Now, we invoke one of the interior axioms, namely . So, by direct computation, we have
The second equality and the inclusion follow from the general properties of the interior operation, the proofs of which can be found here (http://planetmath.org/DerivationOfPropertiesOnInteriorOperation). ∎
Remark. The fact that is open is essential. Otherwise, the proposition fails in general. For example, the rationals , as a subset of the reals under the usual order topology, is not open, and its boundary is not nowhere dense, as , whose interior is itself, and thus not empty.
|Title||boundary of an open set is nowhere dense|
|Date of creation||2013-03-22 17:55:41|
|Last modified on||2013-03-22 17:55:41|
|Last modified by||CWoo (3771)|