Path-connectedness of the fiber ine the LSAH spectral sequence

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I've read some proofs of the identification of the second term of the Leray-Serre-Atiyah-Hirzebruch spectral sequence, and I don't quite seem to understand where in the proof do we need the fiber of the fibration of interest to be path-connected? Moreover, are there `counter-examples´, when the fiber is not path-connected?