The concept of C k-spaces is introduced, situated at an intermediate stage between H-spaces and T-spaces. The C k-space corresponds to the k-th Milnor-Stasheff filtration on spaces. It is proved that a space X is a C k-space if and only if the Gottlieb set G(Z, X) = [Z, X] for any space Z with cat Z ≤ k, which generalizes the fact that X is a T-space if and only if G(σB, X) = [σB, X] for any space B. Some results on the C k-space are generalized to the C k-space for a map f : A → X. Projective spaces, lens spaces and spaces with a few cells are studied as examples ofC k-spaces, and non-C k-spaces.
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