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closed set∗
yark†
2013-03-21 13:49:08
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.
Closed subsets can also be characterized as follows:
A subset C ⊆ X is closed if and only if C contains all of its cluster points,
that is, C 0 ⊆ 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.
∗ hClosedSeti created: h2013-03-21i by: hyarki version: h32739i Privacy setting: h1i
hDefinitioni h54-00i
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