Can you give the definition of:

Closed Set.

Closure of a set.

Open set

Interior of a set?

Boundary of a set.

Isolated Point.

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.