open set in contains an open rectangle
Theorem Suppose is equipped with the usual topology induced by the open balls of the Euclidean metric. Then, if is a non-empty open set in , there exist real numbers for such that and is a subset of .
Proof. Since is non-empty, there exists some point in . Further, since is a topological space, is contained in some open set. Since the topology has a basis consisting of open balls, there exists a and such that is contained in the open ball . Let us now set and for all . Then can be parametrized as
For an arbitrary point in , we have
so , and the claim follows.
|Title||open set in contains an open rectangle|
|Date of creation||2013-03-22 14:07:46|
|Last modified on||2013-03-22 14:07:46|
|Last modified by||matte (1858)|