Closed set with rationals question

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In summary, the question is asking to prove that if a set A contains all rational numbers in the closed interval [0,1], then [0,1] is a subset of A. The confusion arises because A, which is the set of all rationals in [0,1], is not closed, even though every point in A is a boundary point. The suggestion is to prove that any irrational number in [0,1] is in the closure of the rationals in [0,1].
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vancouver_water
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Homework Statement


If A is a closed set that contains every rational number in the closed interval [0,1], show that [0,1] is a subset of A.

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The Attempt at a Solution


I'm confused because for the set A = all rationals in [0,1], every point is a boundary point so the set is closed. but clearly [0,1] is not a subset of A. This question is from Spivaks Calculus on Manifolds, question 1-19
 
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vancouver_water said:
every point is a boundary point so the set is closed
Though the boundary of a set is closed, that doesn't mean that a set A consisting only of boundary points is necessarily closed (the boundary of A could contain points outside of A too).
The set of all rationals in [0,1] is not closed.

You could try to prove that any (irrational) number in [0,1] is in the closure of the rationals in [0,1].
 
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oh right I didn't realize the rationals was not closed, I think I got it now, Thanks!
 

FAQ: Closed set with rationals question

What is a closed set with rationals?

A closed set with rationals is a set of numbers that contains all rational numbers within a given interval. It is called closed because it includes its boundary points and does not have any gaps or holes.

How is a closed set with rationals different from an open set with rationals?

A closed set with rationals includes its boundary points, while an open set with rationals does not. In other words, a closed set with rationals is a closed interval, while an open set with rationals is an open interval.

How do you determine if a set with rationals is closed?

To determine if a set with rationals is closed, you can check if it contains all its boundary points. You can also use the closure property, which states that the closure of a set is the smallest closed set that contains all the points in the original set.

Can a set with rationals be both open and closed?

No, a set with rationals cannot be both open and closed. It can either be open or closed, but not both at the same time. However, a set with rationals can be neither open nor closed.

What is the importance of closed sets with rationals in mathematics?

Closed sets with rationals are important in mathematics because they provide a way to define and work with intervals in a more precise and formal manner. They also have many applications in real analysis, topology, and other areas of mathematics.

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