T-duality as permutation of coordinates in double space

We introduce the 2D dimensional double space with the coordinates Z(M) = (x(mu),y(mu),), whose components are the coordinates of initial space xtk and its T-dual y(mu),. We shall show that in this extended space the T-duality transformations can be realized simply by exchanging the places of some coordinates x(a), along which we want to perform T-duality, and the corresponding dual coordinates y(a). In such an approach it is evident that T-duality leads to the physically equivalent theory and that a complete set of T-duality transformations forms a subgroup of the 2D permutation group. So, in double space we are able to represent the backgrounds of all T-dual theories in a unified manner.