I think it should be Should I open the window. But my grammar book says that I am wrong.
I thought about my question again and realized what I wanted to say was that Shall I open the window, as a sentence, doesn't seem to make sense. This sentence is a question that can't be answered. How can another person know whether I will open the window or not?
Shall I is an expression used in English to volunteer to do something. It's more or less like asking would you like me to open the window?
In my humble opinion "should I" sounds a bit more tentative, as though asking for permission, almost.
not a teacher
Hedwig's explanation is how I would use them in BrE.