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

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