A=(0,2) B=(1,3)

A and B are open.

The boundary points of A are 0 and 2, neither of which belong to A

The boundary points of B are 1 and 3, neither of which belong to B

A^B = (1,2)

A=(0,2) B=[1,3)

The boundary point 1 of B belongs to B

A^B = [1,2), The interior is (1,2)

The boundary point 1 in B is also a point of A so it belongs to the intersection A^B, where it is also a boundary point because to the right of 1 are points common to A and B and to the left of 1 are only points of A.

A=(0,2] B = [1,3)

A^B = [1,2]. The interior is (1,2)

P is an interior point of A if it has a neighborhood entirely in A .