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If $f:E\to B$ is a Fiber Bundle with $B$ a Paracompact Topological Space, then $f$ satisfies the Homotopy Lifting Property with respect to all Topological Spaces. In other words, if $g: [0,1] \times X\to B$ is a Homotopy from $g_0$ to $g_1$, and if $g'_0$ is a Lift of the Map $g_0$ with respect to $f$, then $g$ has a Lift to a Map $g'$ with respect to $f$. Therefore, if you have a Homotopy of a Map into $B$, 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 $f:E\to B$ such that it satisfies the Homotopy Lifting Property.

See also Fiber Bundle, Fiber Space

© 1996-9 Eric W. Weisstein