proof that a compact set in a Hausdorff space is closed
Let be a Hausdorff space, and a compact subset. We are to show that is closed. We will do so, by showing that the complement is open. To prove that is open, it suffices to demonstrate that, for each , there exists an open set with and .
For any point , we have , and therefore for some . Since and are disjoint, , and therefore . Thus is disjoint from , and is contained in .
|Title||proof that a compact set in a Hausdorff space is closed|
|Date of creation||2013-03-22 13:34:54|
|Last modified on||2013-03-22 13:34:54|
|Last modified by||yark (2760)|