Corrected for grammar and conciseness: Provide an example of each of the following, or explain why you believe such a set cannot exist: (a) A nonempty set without accumulation points or isolated points. (b) A nonempty set without interior points or isolated points. (c) A nonempty set without boundary points or isolated points.

(a) A nonempty set without accumulation points or isolated points cannot exist. This is because by definition, an accumulation point (or limit point) of a set is a point where every neighborhood of it contains at least one point of the set different from the point itself. If a set does not have accumulation points, then each of its points must be isolated; there will be a neighborhood around each point that contains no other points of the set. Therefore, a set cannot be nonempty and have neither accumulation points nor isolated points.

(b) A nonempty set without interior points or isolated points can exist. For example, the set of rational numbers \(\mathbb{Q}\) within the interval [0,1] is such a set. This set does not have any interior points because any interval you take will contain both rational and irrational numbers; thus, there is no interval around a point that is composed entirely of rational numbers. Likewise, there are no isolated points because between any two rational numbers there is another rational number – they are densely packed.

(c) A nonempty set without boundary points or isolated points cannot exist. A boundary point of a set is a point where every neighborhood of it contains at least one point of the set and at least one point not in the set. If a set has no boundary points, then no points would be approachable from outside the set, which effectively means the set would have to be the entire space. However, if the set were the entire space, it could not have isolated points, as every point would be surrounded by other points of the set. Since we need a nonempty set that isn't the whole space, for practical purposes in a standard topological space, such a set cannot exist.