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AxiomOfChoice
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The set [itex]\{1,2,3,4,5\ldots\}[/itex]...is it closed as a subset of [itex]\mathbb{R}[/itex]? I'm thinking "yes," but I'm unsure of myself for some reason. (And yes, this is just the set of positive integers.
Yes. Any union of open sets is open. Thanks! :)Office_Shredder said:The definition of closed is the complement of an open set. So is the set
[tex]( - \infty,1) \cup (1,2) \cup (2,3) \cup (3,4)...[/tex] open?
Yes, the set {1,2,3,4,5...} is a closed set in ℝ. A closed set in ℝ is a set that contains all of its limit points. In this set, the limit points are the numbers that are larger than or equal to 1. Therefore, the set is closed as it contains all of its limit points.
A closed set in ℝ is a set that contains all of its limit points. A limit point is a number that can be approached arbitrarily close to by a sequence of numbers in the set. In other words, for a set to be closed in ℝ, it must include all of its boundary points.
A closed set in ℝ includes all of its limit points, while an open set does not include its boundary points. This means that a closed set is "closed off" from its surroundings, while an open set is "open" to its surroundings. Another way to think about it is that a closed set contains all of its possible values, while an open set does not.
No, a set cannot be both closed and open in ℝ. This is because if a set is closed, it contains all of its limit points and therefore cannot have any points "missing" that would make it open. Similarly, if a set is open, it does not contain its boundary points and therefore cannot be closed.
Some examples of closed sets in ℝ include the set of all integers (ℤ), the set of all non-negative real numbers [0, ∞), and the set of all real numbers (ℝ) itself. These sets all contain all of their limit points and are therefore closed in ℝ.