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If is a Fiber Bundle with
a Paracompact Topological Space,
then
satisfies the Homotopy Lifting Property with respect to all Topological
Spaces. In other words, if
is a Homotopy from
to
, and if
is a Lift of the Map
with respect to
, then
has a Lift to a Map
with
respect to
. Therefore, if you have a Homotopy of a Map into
, and if the beginning of it has a
Lift, then that Lift can be extended to a Lift of the Homotopy itself.
A fibration is a Map between Topological Spaces such that it satisfies
the Homotopy Lifting Property.
See also Fiber Bundle, Fiber Space