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 $H^- = H = \openint a b$.
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Examples $3.7.14 \ \text {(b)}$