closed set (original) (raw)

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closed under

Let (X,τ) be a topological space. Then a subset C⊆X is closed if its complement X∖C is open under the topology τ.

Examples:

• •
  In any topological space (X,τ), the sets X and ∅ are always closed.
• •
  Consider ℝ with the standard topology. Then [0,1] is closed since its complement (-∞,0)∪(1,∞) is open (being the union of two open sets).
• •
  Consider ℝ with the lower limit topology. Then [0,1) is closed since its complement (-∞,0)∪[1,∞) is open.

A subset C⊆X is closed if and only if C contains all of its cluster points, that is, C′⊆C.

So the set {1,1/2,1/3,1/4,…} is not closed under the standard topology on ℝ since 0 is a cluster point not contained in the set.