Boundary (Topology)/Examples/Rationals in Closed Unit Interval

Examples of Boundaries in the context of Topology

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 $\closedint 0 1$.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: Exercise $3.9: 31$