example of a space that is not semilocally simply connected
An example of a space that is not semilocally simply connected is the following: Let
endowed with the subspace topology. Then has no simply connected neighborhood. Indeed every neighborhood of contains (ever diminshing) homotopically non-trivial loops. Furthermore these loops are homotopically non-trivial even when considered as loops in .
The Hawaiian rings
It is essential in this example that is endowed with the topology induced by its inclusion in the plane. In contrast, the same set endowed with the CW topology is just a bouquet of countably many circles and (as any CW complex) it is semilocaly simply connected.
|Title||example of a space that is not semilocally simply connected|
|Date of creation||2013-03-22 13:25:10|
|Last modified on||2013-03-22 13:25:10|
|Last modified by||mathcam (2727)|