is path connected if is countable
We use simply as an example; an analogous proof will work for any .
Fix a point not in . The strategy of the proof is to construct a path from any to . If we can do this then for any we may construct a path from to by first following and then following in reverse.
Fix , and consider the set of all (straight) lines through . There are uncountably many of these and they meet in the single point , so not all of them contain a point of . Choose one that doesn’t and move along it: your distance from takes on uncountably many values, and hence at some point this distance from is not shared by any point of . The whole of the circle with radius , centre , lies in so we may move around it freely.
|Title||is path connected if is countable|
|Date of creation||2013-03-22 16:09:11|
|Last modified on||2013-03-22 16:09:11|
|Last modified by||silverfish (6603)|