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I would like to trow away some confusion I'm accumulating.

  1. Usually when superstrings are introducted (see for example Wikipedia) one says: Type I is a theory of open and closed strings, while Type II (A and B) is a theory of only closed strings. That is ok;
  2. After some exploration ones find that one of most important object in Type II theory are Dbrane, where, by definition, open strings can end. So in Type II theory there are open string. As you can see at page 53 of the PDF, Tong affirm:

For type II superstrings, the open strings and Dbranes are necessary ingredients.

How can the apparent tension (if not contraddiction) of points 1. 2. be solved?

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I think the idea is that there are no open strings in IIA, IIB in perturbation theory around the vacuum. As you say, we know D-branes exist in IIA and IIB, and these are defined as the submanifolds where open strings can end, so if a theory has D-branes it should also have open strings. But to have an open string means that you necessarily have a heavy, solitonic-like object (the D-brane), the D-brane tension goes like $T_p \propto 1/g_s$, so if $g_s \ll 1$ as it would be in perturbation theory, the D-branes are non-perturbative objects. Therefore you would not say that IIA, IIB has open strings in perturbation theory around the vacuum; they have open strings in perturbation theory around the various D-brane solutions.

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