Question about open and closed sets

Hi guys,

I have two questions wanting to clarify.

1) My lecture notes say that if a set is open then its complement is closed. However some books also say that the converse is true: that is if a set is closed, then its complement is open. Is this correct?

2) Is a boundary point a limit point? The definition says that a boundary point x which belongs to S is a point if, for all epsilon > 0, the neighborhood of x contains both S and the complement of S. Doesn't this make x a limit point also? Since you can approach x from elements in the set S.

Thanks for reading.