testing continuity via filters
Suppose first is continuous. Let be a filter in converging to . We want to show that converges to . Let be a neighborhood of . So there is an open set such that . So is open and contains , which means that by assumption. This means that . Since , we see that as well.
Conversely, suppose preserves converging filters. Let be an open set in containing . We want to find an open set in containing , such that . Let be the neighborhood filter of . So . By assumption, . Since is an open neighborhood of , we have , or for some . Since is a neighborhood of , it contains an open neighborhood of . Furthermore, . Since is arbitrary, is continuous. ∎
|Title||testing continuity via filters|
|Date of creation||2013-03-22 19:09:31|
|Last modified on||2013-03-22 19:09:31|
|Last modified by||CWoo (3771)|