Accumulation Points of Rationals: Explained

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SUMMARY

The accumulation points of rational numbers are all real numbers, as established by the density property of rationals, which states that between any two real numbers, there exists a rational number. The discussion clarifies that the set of rational numbers, denoted as Q, is neither open nor closed in the real number space R. The definition of an accumulation point is provided, stating that a point x is an accumulation point of a set A if any open set U containing x intersects with A\{x} and is not empty. The derived set of A, denoted A', contains all accumulation points of A.

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  • Familiarity with rational numbers and their density
  • Knowledge of open and closed sets in topology
  • Basic concepts of limit points and derived sets
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Accumulation points of rationals and open or closed.

I know the accumulation points are all real but I don't understand why.

The set is neither open nor closed to but I don't truly see it.

Can someone explain both?
 
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dwsmith said:
Accumulation points of rationals and open or closed.

I know the accumulation points are all real but I don't understand why.

The set is neither open nor closed to but I don't truly see it.

Can someone explain both?

The definition of limit point or accumulate point, x is accumulate point of a set A if any open set U containing x, A\{x} intersect with U is not empty.

Let x in R any open set contains x will intersect with Q since the density property of Q which says between any two real numbers there exist a rational, that holds for any x real so the accumulate point of Q is R.

the set which contains all accumulate point of A called the derive set A'

what do you mean by "The set is neither open nor closed to but I don't truly see it." which set are you taking about ?
 

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