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We define and study a concept of T f -space for a map, which is a generalized one of a T -space, in terms of the Gottlieb set for a map. We show that X is a T f -space if and only if G(ΣB; A, f, X) = [ΣB, X] for any space B. For a principal fibration Ek → X induced by k : X → X ' from (cid:178) : P X ' → X ', we ¯f -structure on Ek obtain a suﬃcient condition to having a lifting T of a T f -structure on X . Also, we define and study a concept of co-T g-space for a map, which is a dual one of T f -space for a map. We obtain a dual result for a principal cofibration ir : X → Cr induced by r : X ' → X from ι : X ' → cX '.
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