  We extend the formalism of Topological T-duality to spaces which are the
total space of a principal $S^1$-bundle $p:E \to W$ with an $H$-flux in
$H^3(E,Z)$ together the together with an automorphism of the continuous-trace
algebra on $E$ determined by $H$. The automorphism is a `topological
approximation' to a gerby gauge transformation of spacetime. We motivate this
physically from Buscher's Rules for T-duality. Using the Equivariant Brauer
Group, we connect this problem to the $C^{\ast}$-algebraic formalism of
Topological T-duality of Mathai and Rosenberg.
  We show that the study of this problem leads to the study of a purely
topological problem, namely, Topological T-duality of triples $(p,b,H)$
consisting of isomorphism classes of a principal circle bundle $p:X \to B$ and
classes $b \in H^2(X,Z)$ and $H \in H^3(X,Z).$ We construct a classifying space
$R_{3,2}$ for triples in a manner similar to the work of Bunke and Schick
\cite{Bunke}. We characterize $R_{3,2}$ up to homotopy and study some of its
properties. We show that it possesses a natural self-map which induces
T-duality for triples. We study some properties of this map.
