A map p:E→B with B not necessarily path-connected is defined to be a quasifibration if the following equivalent conditions are satisfied:
(i) For all b∈B and x0∈p−1(b), the map p∗:πi(E,p−1(b),x0)→πi(B,b) is an isomorphism for i>0 and π0(p−1(b),x0)→π0(E,x0)→π0(B,b) is exact.
(ii) The inclusion of the fiber p−1(b) into the homotopy fiber Fb of p over b is a weak homotopy equivalence for all b∈B.
(iii) The restriction of p over each path-component of B is a quasifibration according to the definition in this section.
Show these three conditions are equivalent, and prove Lemma 4K.3 for quasifibrations over non-path-connected base spaces.