# Equivalence of Definitions of Boundary

## Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $H \subseteq S$.

The following definitions of the concept of **boundary of $H$** are equivalent:

### Definition $1$: from Closure and Interior

The **boundary of $H$** consists of all the points in the closure of $H$ which are not in the interior of $H$.

Thus, the **boundary of $H$** is defined as:

- $\partial H := H^- \setminus H^\circ$

where $H^-$ denotes the closure and $H^\circ$ the interior of $H$.

### Definition $2$: from Neighborhood

$x \in S$ is a **boundary point** of $H$ if every neighborhood $N$ of $x$ satisfies:

- $H \cap N \ne \O$

and

- $\overline H \cap N \ne \O$

where $\overline H$ is the complement of $H$ in $S$.

The **boundary of $H$** consists of all the **boundary points** of $H$.

### Definition $3$: from Intersection of Closure with Closure of Complement

The **boundary of $H$** is the intersection of the closure of $H$ with the closure of the complement of $H$ in $T$:

- $\partial H = H^- \cap \paren {\overline H}^-$

### Definition $4$: from Closure and Exterior

The **boundary of $H$** consists of all the points in $H$ which are not in either the interior or exterior of $H$.

Thus, the **boundary of $H$** is defined as:

- $\partial H := H \setminus \paren {H^\circ \cup H^e}$

where:

## Proof

### Definition $1$ is equivalent to Definition $3$

This is demonstrated in Boundary is Intersection of Closure with Closure of Complement.

$\Box$

### Definition $2$ is equivalent to Definition $3$

Let $x \in S$.

By definition of the closure:

- $x \in H^-$ if and only if every neighborhood $N$ of $x$ satisfies $H \cap N \ne \O$
- $x \in \paren{\overline H}^-$ if and only if every neighborhood $N$ of $x$ satisfies $\overline H \cap N \ne \O$

Therefore $x \in H^- \cap \paren{\overline H}^-$ if and only if every neighborhood $N$ of $x$ satisfies:

- $H \cap N \ne \O$

and

- $\overline H \cap N \ne \O$

$\Box$

This needs considerable tedious hard slog to complete it.In particular: Definition 4To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |