T-duality in string theory
We offer new interpretation of T-dualization procedure of the bosonic and of the type IIB superstring theory in double space framework. We introduce the $2D$ dimensional double space with the coordinates $Z^M= (x^\mu, y_\mu)$ which components are the coordinates of initial space $x^\mu$ and its T-dual $y_\mu$. In this extended space the T-duality transformations are equivalent to the permutation of some coordinates $x^a$, along which we want to perform T-duality and the corresponding dual coordinates $y_a$. In such approach it is evident that T-duality leads to the physically equivalent theory.
So, in the double space we are able to represent the backgrounds of all T-dual theories in unified manner. For example, in supersymmetric case such formulation describes both type IIB and type IIA superstring theories. Exchanging odd number of coordinate with the corresponding dual ones we can switch from type IIB to type IIA theory.