Closure (Topology)/Examples
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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:
- $H^- = \openint a b$