Definitions#

Interior Point $x \in C$
A point is an interior point if we can find a ball around x which is completely inside C.
Open Set
A set is open, if all points are interior points.
Closed Set
A set is closed, if its complement is open.
Closure
Take the set’s complement. Find its interior. Take the complement of it. You got a closure.

A point is in a closure, if for every $\epsilon$, you can find a point from the original set within $\epsilon$ distance.

Boundary
take the closure and remove the interior.

Examples#

For example, closure of (-1, 1) in real line is given by complement of the interior of the (-inf, 1] U [1, inf). Which is the complement of (-inf, 1) U (1, inf), which is [-1, 1].