limit points and closure for connected sets
Suppose is a connected set in a topological space. If , then is connected. In particular, is connected.
Thus, one way to prove that a space is connected is to find a dense subspace in which is connected.
Let be the ambient topological space. By assumption, if are open and , then . To prove that is connected, let be open sets in such that and for a contradition, suppose that . Then there are open sets such that
It follows that and . Next, let be open sets in defined as
and as , it follows that . Then, by the properties of the closure operator,
|Title||limit points and closure for connected sets|
|Date of creation||2013-03-22 15:17:56|
|Last modified on||2013-03-22 15:17:56|
|Last modified by||matte (1858)|