Definition:Closed Set/Metric Space/Definition 1
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Definition
Let $M = \struct {A, d}$ be a metric space.
Let $H \subseteq A$.
$H$ is closed (in $M$) if and only if its complement $A \setminus H$ is open in $M$.
Also see
- Results about closed sets can be found here.
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Compactness
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 6$: Open Sets and Closed Sets: Definition $6.5$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): closed set
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis: Chapter $1$: Normed and Banach spaces