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 $\closedint 0 1$.