Limit Point/Examples/Union of Singleton with Open Real Interval
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Examples of Limit Points
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 $0$ is not a limit point of $H$.
Proof
Consider the open $1$- ball $\map {B_1} 0$ of $0$.
Then the only element of $H$ which is in $\map {B_1} 0$ is $0$ itself.
Hence $0$ is not a limit point of $H$ by definition.
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Definition $3.7.10 \ \text {(b)}$