The compactified long closed ray overlineR will have two endpoints,
but these are distinguishable. One has a neighbourhood
homeomorphic to [0,1) and the other doesn't. This scuppers
"long homotopy" being a symmetric relation.
(Also the transitivity would fail too.)
The standard notion of homotopy relies on the interval I
having distinguished points 0 and 1, there being a self
map of I swapping 0 and 1, and there being a map from
IcoprodI/sim to I where sim is the equivalence relation
identifying the 1 in the first component to the 0 in the second.
These maps have to satisfy various formal properties. There
is no continuous map of overlineR swapping its "endpoints",
so we can't mimic the classical notion of homotopy.
No comments:
Post a Comment