The set of interior points of R is not empty

Definition of Interior Points
The interior of a set S is defined as the set of all points in S that do not lie on the boundary of S. In other words, a point x is an interior point of S if there exists a neighborhood of x that is contained entirely within S.

Explanation
To determine whether the set of interior points of R (the set of real numbers) is empty or not, we need to consider the definition of interior points.

Interior Points of N (Natural Numbers)
The set of natural numbers, N = {1, 2, 3, ...}, does not contain any interior points. This is because for any natural number n, there is no open interval (a, b) such that a < n="" />< b="" entirely="" lies="" within="" n.="" therefore,="" option="" a="" is="" />

Interior Points of I (Irrational Numbers)
The set of irrational numbers, I, is the complement of the set of rational numbers in R. Since the set of rational numbers has no interior points (as every rational number has both rational and irrational numbers on either side of it), the set of irrational numbers also has no interior points. Therefore, option B is correct.

Interior Points of Z (Integers)
The set of integers, Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}, does not contain any interior points. Similar to the set of natural numbers, for any integer n, there is no open interval (a, b) such that a < n="" />< b="" entirely="" lies="" within="" z.="" therefore,="" option="" c="" is="" />

Interior Points of R (Real Numbers)
Unlike the sets of natural numbers, irrational numbers, and integers, the set of real numbers, R, contains interior points. This is because for any real number x, we can always find an open interval (a, b) such that a < x="" />< b,="" and="" this="" interval="" lies="" entirely="" within="" r.="" therefore,="" option="" d="" is="" not="" />

Conclusion
Among the given options, the statement that the set of interior points of R is empty (option D) is not correct. The set of real numbers does contain interior points, unlike the sets of natural numbers, irrational numbers, and integers.