The compactified long closed ray $overline R$ 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
$Icoprod I/sim$ 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 $overline R$ swapping its "endpoints",
so we can't mimic the classical notion of homotopy.
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