Do Nonempty Perfect Sets in R Exist Without Rational Numbers?

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In summary, a perfect set is a closed, uncountable set where every point is an accumulation point. It differs from a closed set in that it also contains all its accumulation points. A perfect set cannot be countable as it must be uncountable to contain all accumulation points. An accumulation point is a point that can be approached closely by other points in the set. While perfect sets have applications in mathematics, they do not have many practical real-life applications.
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AbelAkil
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Is there any nonempty perfect set in R which contains no rational number?
I cannot figure it out...:frown:
I appreciate your solutions!

PS: A set E is perfect iff E is closed and every point of E is a limit point of E
 
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FAQ: Do Nonempty Perfect Sets in R Exist Without Rational Numbers?

What is a perfect set?

A perfect set is a set of numbers that is closed, uncountable, and every point in the set is an accumulation point.

How is a perfect set different from a closed set?

A closed set is a set that contains all its limit points, while a perfect set is a closed set that also contains all its accumulation points.

Can a perfect set be countable?

No, a perfect set cannot be countable because it must be uncountable in order to contain all its accumulation points.

What is an accumulation point?

An accumulation point is a point in a set that can be approached arbitrarily closely by other points in the set.

Are there any real-life applications of perfect sets?

Perfect sets have applications in mathematics, such as in topology and measure theory, but they do not have many practical real-life applications.

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