the topologist’s sine curve has the fixed point property
Let and , and .
If is a continuous map, then since and are both path connected, the image of each one of them must be entirely contained in another of them.
So the only case that remains is that . And since is compact, its projection to the first coordinate is also compact so that it must be an interval with . Thus is contained in . But is homeomorphic to a closed interval, so that it has the fixed point property, and the restriction of to is a continuous map , so that it has a fixed point.
This proof is due to Koro.
|Title||the topologist’s sine curve has the fixed point property|
|Date of creation||2013-03-22 16:59:37|
|Last modified on||2013-03-22 16:59:37|
|Last modified by||Mathprof (13753)|