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In the paper [1] we showed that in double space, where all initialcoordinates $x^\mu$ are doubled $x^\mu \to y_\mu$, the T-dualitytransformations can be performed by exchanging places of some coordinates $x^a$and corresponding dual coordinates $y_a$. Here we generalize this result to thecase of weakly curved background where in addition to the extended coordinatewe will also transform extended argument of background fields with the sameoperator $\hat {\cal T}^a$. So, in the weakly curved background T-duality leadsto the physically equivalent theory and complete set of T-dualitytransformations form the same group as in the flat background. Therefore, thedouble space represent all T-dual theories in unified manner.

T-duality as coordinates permutation in double space for weakly curved background