Two-dimensional Yang-Mills theory at large N has a dual description as a theory of closed strings, making it a useful model for entanglement in gauge/string duality. Definition of entanglement in this theory requires the introduction of an extended Hilbert space, and I will show how this extension fits into the framework of open/closed topological field theory. In the string description, the entanglement entropy counts states of open strings ending on a pointlike object we call an entanglement brane. In the large N limit, the entanglement entropy is of order N^2 and is equal to the expectation value of a local operator, analogous to the Ryu-Takayanagi formula. Intriguingly, the subleading correction to this result grows linearly with N, which is larger than expected from the closed string expansion. This large correction arises from analytic continuation of the spacetime topology and may have implications for entanglement of string theory in higher dimensions.