Boundary (Topology)/Examples

Examples of Boundaries in the context of Topology

Half-Open Real Interval

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Let $\hointl a b$ be a half-open interval of $\R$.

Then the boundary of $\hointl a b$ is the set of its endpoints $\set {a, b}$.

Open Unit Interval

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Let $\openint 0 1$ be the open unit interval in $\R$.

Then the boundary of $\openint 0 1$ is the set of its endpoints $\set {0, 1}$.

$\Z$ in $\R$

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Let $\Z$ be the set of integers.

Then the boundary of $\Z$ in $\struct {\R, \tau_d}$ is $\Z$ itself.

Reciprocals in $\R$

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Let $S$ be the set defined as:

$S = \set {\dfrac 1 n: n \in \Z_{>0} }$

Then the boundary of $S$ in $\struct {\R, \tau_d}$ is $S \cup \set 0$.

Rationals in Closed Unit Interval in $\R$

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Let $S$ be the set defined as:

$S = \Q \cap \closedint 0 1$

where:

$\Q$ denotes the set of rational numbers

$\closedint 0 1$ denotes the closed unit interval.

Then the boundary of $S$ in $\struct {\R, \tau_d}$ is $\closedint 0 1$.