Closure, interior and boundary

These three things have their own page because of how often they come together.

Closure

The closure of a set $A$ denoted $\bar{A}$ ("bar" in LaTeX) or $\overline{A}$ ("overline" in LaTeX) is the set:

$$\overline{A}=\bigcap\{B\subset X|A\subset B\text{ and }B\text{ is closed in }X\}$$

Alternatives

Alternatively if $$A'$$ denotes the set of all limit points of $$A$$ then the closure can be defined as:

$$\bar{A}=A\cup A'$$

Interior

The interior of $A$ denoted by $\text{Int }A$ or $\text{Int}(A)$ is the set:

$$\text{Int}(A)=\bigcup\{C\subset X|C\subset A\text{ and }C\text{ is open in }X\}$$

Exterior

A less common but still very useful notion is that of exterior, denoted $\text{Ext }A$ or $\text{Ext}(A)$ given by:

$$\text{Ext}(A)=X-\overline{A}$$

Boundary

The boundary of $A$, denoted by $\partial A$ is given by:

$$\partial A=X-(\text{Int}(A)\cup\text{Ext}(A))$$