Closure (Topology)/Examples

Examples of Closure in the context of Topology

Singleton Union with Open Interval

Let $\R$ be the set of real numbers.

Let $H \subseteq \R$ be the subset of $\R$ defined as:

$H = \set 0 \cup \openint 1 2$

Then the closure of $H$ in $\R$ is:

$H^- = \set 0 \cup \closedint 1 2$

Open Interval in Open Unbounded Interval

Let $S$ be the open real interval:

$S = \openint a \to$

Let $H$ be the open real interval:

$H = \openint a b$

Then the closure of $H$ in $S$ is:

$H^- = \hointl a b$

Open Interval under Discrete Topology

Let $\T = \struct {\R, \tau_d}$ denote the topological space formed from the set of real numbers $\R$ together with the discrete topology $\tau_d$.

Let $H$ be the open real interval:

$H = \openint a b$

Then the closure of $H$ in $T$ is:

$H^- = \openint a b$