  In this paper, we prove that the open and closed strings are $O(D,D)$
equivalent. The equivalence requires an AdS geometry near the boundaries. The
$O(D,D)$ invariance is introduced into the Polyakov action by the Tseytlin's
action. Traditionally, there exist disconnected open-open or closed-closed
configurations in the solution space of the Tseytlin's action. The open-closed
configuration is ruled out by the mixed terms of the dual fields. We show that,
under some very general guidances, the dual fields are consistently decoupled
if and only if the near horizon geometry is $AdS_5$. We then have open-closed
and closed-open configurations in different limits of the distances of the
$D3$-brane pairs. Inherited from the definition of the theory, these four
configurations are of course related to each other by $O(D,D)$ transformations.
We therefore conclude that both the open/closed relation and open/closed
duality can be derived from $O(D,D)$ symmetries. We then demonstrate the
open/closed relation does connect commutative open and closed strings. By
analyzing the couplings of the configurations, the low energy effective limits
of our results consequently predicts the AdS/CFT correspondence, Higher spin
theory, weak gauge/weak gravity duality and a yet to be proposed strong
gauge/strong gravity duality. Furthermore, we also have the Seiberg duality and
a weak/strong gravitation duality as consequences of $O(D,D)$ symmetries.
