Definition:Closed Set/Metric Space/Definition 1

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

• Equivalence of Definitions of Closed Set in Metric Space

• 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