Answer:

To understand why the correct answer is option 'A', let's break down the concepts involved and explain them in detail.

1. Bounded Below:
A set of real numbers is said to be bounded below if there exists a real number that is less than or equal to every element in the set. In other words, there is a lower bound for the set.

2. Infimum (inf):
The infimum of a set S of real numbers is defined as the greatest lower bound of S. It is denoted by inf S. Essentially, the infimum is the smallest real number that is greater than or equal to every element in the set.

3. Point of Closure:
A point of closure to a set S is a point that is either an element of S or a limit point of S. A limit point of a set S is a point such that every neighborhood of the point contains infinitely many points of S.

Explanation:

Now, let's consider a set S of real numbers that is bounded below and analyze the options given.

a) A point of closure to S:
The infimum of S, inf S, is the greatest lower bound of S. Since S is bounded below, there exists a real number that is less than or equal to every element in S. Therefore, inf S is a point that is less than or equal to every element in S, making it a point of closure to S.

b) Not a point of closure to S:
This option is incorrect because, as explained above, the infimum of S is indeed a point of closure to S.

c) Prime number:
The concept of prime numbers is unrelated to the question and does not provide any information about the infimum of a set of real numbers.

d) None of the above:
This option is incorrect because, as explained above, option 'A' is the correct answer.

Therefore, the correct answer is option 'A' - The infimum of S is a point of closure to S.