It is shown that all the generalized Whitehead products vanish in X and all the components of X TA have the same homotopy type when X is a T-space. It is also shown that any T-space is a G-space. The dual spaces of T-spaces are introduced and studied.1991 Mathematics subject classification (Amer. Math. Soc): 55P45, 55P35.

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In this paper, we introduce and study the concepts of weak C f k-spaces for maps which are generalized concepts of C f kspaces for maps, and introduce the dual concepts of weak C f k-spaces for maps and obtain some dual results.