The open mapping theorem states that a bounded surjective linear operator T on a Complete normed space X, mapping it to a Complete normed linear space Y, will be an open map.

My argument is that one can prove this statement without the fact that the spaces are complete!

What follows from this theorem is that a bijective linear operator on complete spaces will have an inverse that is bounded. This I can also prove without needing completeness.

Am I missing a point here? Why is it always required that the spaces are complete, in order for the theorem to hold?