Closure of the Rational Numbers (Using Standard Topology)

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Homework Help Overview

The discussion centers around the closure of the rational numbers in the context of standard topology, specifically addressing the proof that Cl(Q) equals R. Participants are navigating the concepts of closure and limit points, which have not yet been covered in their coursework.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are discussing the definition of closure and considering the implications of closed sets containing the rational numbers. There is an emphasis on understanding the complement of closed sets and the presence of both rational and irrational numbers within any interval of real numbers.

Discussion Status

The discussion is active, with participants questioning definitions and exploring the properties of closed sets. Some guidance has been offered regarding the description of closed sets and their complements, but no consensus has been reached on the proof itself.

Contextual Notes

Participants note that limit points have not been covered in their studies, which may impact their ability to approach the proof effectively. There is an underlying assumption that understanding the closure concept is essential for this problem.

Fluffman4
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Prove that Cl(Q) = R in the standard topology
I'm really stuck on this problem, seeing as we haven't covered limit points yet in the text and are not able to use them for this proof. Can anybody provide me with help needed for this proof? Many thanks.
 
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You should state the definition of 'Cl(Q)' you are using. Is it the intersection of all closed sets containing Q? If so, think about what the complement of a closed set containing Q must be.
 
We defined the closure of a subset A of a topological space as the intersection of all closed sets containing A. We basically want to find the smallest closed set containing A for closure.
 
Fluffman4 said:
We defined the closure of a subset A of a topological space as the intersection of all closed sets containing A. We basically want to find the smallest closed set containing A for closure.

Ok. So describe a closed set A in R containing Q. You might find it easier to describe the complement of A.
 
Crucial point: given any interval of real numbers, there exist both rational and irrational numbers in that interval.
 

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