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

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$x \in S$ is a boundary point of $H$ if and only 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 point 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:

$H^\circ$ denotes the interior of $H$

$H^e$ denotes the exterior of $H$.

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

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

$\Box$

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