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?
Accumulation points of rationals, also known as limit points, are points on the real number line that are approached infinitely closely by a sequence of rational numbers.
Unlike isolated points, accumulation points have an infinite number of rational numbers approaching them. They are also distinct from endpoints, as they are not finite numbers on the number line.
Accumulation points of rationals are closely related to the concept of limits in calculus. In fact, they can be thought of as the "building blocks" of limits, as they represent the points where a function approaches a certain value.
Yes, accumulation points of rationals can be irrational numbers. This is because they are defined as points on the real number line, which includes both rationals and irrationals.
Understanding accumulation points of rationals can be useful in various mathematical applications, such as in the study of limits, continuity, and sequences. It also helps to provide a more complete understanding of the real number line, and the relationships between different types of numbers.