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

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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 $H^- = H = \openint a b$.

$\blacksquare$


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