Closure (Metric Space)/Examples/Union of Disjoint Closed Real Intervals
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Examples of Closure in the context of a Metric Space
Let $\R$ be the real number line under the usual (Euclidean) metric.
Let $M$ be the subspace of $\R$ defined as:
- $M = \closedint 0 1 \cup \closedint 2 3$
Let $\map {B_1} 1$ denote the open $1$-ball of $1$ in $M$.
Let $\map { {B_1}^-} 1$ denote the closed $1$-ball of $1$ in $M$.
Then:
- $\map \cl {\map {B_1} 1} = \closedint 0 1$
while:
- $\map { {B_1}^-} 1 = \closedint 0 1 \cup \set 2$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Example $3.7.22 \ \text {(b)}$