Closure (Topology)/Examples/Open Interval under Discrete Topology

Example of Closure in the context of 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:
 * $H^- = \openint a b$

Proof
From Set in Discrete Topology is Clopen, $H = \openint a b$ is both open and closed.

From Set is Closed iff Equals Topological Closure it follows that $H^- = H = \openint a b$.