proof that components of open sets in a locally connected space are open
First, suppose that is locally connected and that is an open set of . Let , where is a component of . Since is locally connected there is an open connected set, say with . Since is a component of it must be that . Hence, is open. For the converse, suppose that each component of each open set is open. Let . Let be an open set containing . Let be the component of which contains . Then is open and connected, so is locally connected.
As a corollary, we have that the components of a locally connected space are both open and closed.
|Title||proof that components of open sets in a locally connected space are open|
|Date of creation||2013-03-22 17:06:07|
|Last modified on||2013-03-22 17:06:07|
|Last modified by||Mathprof (13753)|