testing for continuity via nets
Conversely, suppose is not continuous, say, at a point . Then there is an open set containing such that does not contain any open set containing . Let be the set of all open sets containing . Then under reverse inclusion, is a directed set (if , then ). Define a relation as follows:
Then for each , there is an such that , since . By the axiom of choice, we get a function from to . Write . Since is directed, is a net. In addition, converges to (just pick any , then for any , we have by the definition of ). However, does not converge to , since for any .
. Suppose nets and both converge to . Then, by assumption, and are nets converging to .
Conversely, suppose converges to , and is indexed by a directed set . Define a net such that for all . Then clearly converges to . Hence both and converge to the same point in . But converges to , we see that converges to as well. ∎
|Title||testing for continuity via nets|
|Date of creation||2013-03-22 19:08:58|
|Last modified on||2013-03-22 19:08:58|
|Last modified by||CWoo (3771)|