MHB Accumulation Points of Rationals: Explained

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Accumulation points of rational numbers are all real numbers due to the density property of rationals, which states that between any two real numbers, there exists a rational number. This means that any open set containing a real number will intersect with the set of rational numbers, confirming that all real numbers are accumulation points. The set of rational numbers is neither open nor closed in the real number topology. The derived set of rational numbers, which includes all accumulation points, is the entire set of real numbers. Understanding these properties clarifies the nature of rational numbers in relation to accumulation points.
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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 ?
 

Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
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