Analysis (Not Rigorous) Notes

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].