Can you give the definition of:
Closure of a set.
Interior of a set?
Boundary of a set.
Can you tell us what tools you have:
To prove a set is closed.
To prove a set is open
To prove a point is an interior point.
To prove a point is a boundary point.
To prove a point is an isolated point.
Until you answer those questions, it is almost impossible to help you.
Definitions vary so much from textbook to textbook.