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at Serre fibration I have spelled out the proof that that with $F_x \hookrightarrow X \overset{fib}{\to} Y$ then $\pi_\bullet(F) \to \pi_\bullet(X)\to \pi_{\bullet(Y)}$ is exact in the middle. here.
(This is intentionally the low-technology proof using nothing but the definition. )
Added to Serre fibration a section Properties – Relation to Hurewicz fibrations with pointer to
a) counter-example of a Serre fibration that is not a Hurewicz fibration
b) statement that all Serre fibrations between CW-complexes are Hurewicz fibrations.
Also added to the Examples-section pointer to the homotopy lifting property for covering spaces.
(The projection map of) a fibre bundle over any paracompact space, or more generally, a bundle that admits a trivialisation over a numerable cover, is a Hurewicz fibration, I believe.. In general, all fibre bundles are Serre fibrations.
(I was torn between adding this comment here or at the covering space thread)
Fixed some syntax errors
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