We can define some sets within the superset of all real numbers:
A. Integers
A1. Positive integers or natural numbers, including zero
A1.1. Positive integers with integer exponents greater than zero.
A1.2. Negative integers with even integer exponents greater than zero
A2. Negative integers
A2.1. Negative integers with odd integer exponents greater than zero
B. Fractions
B1. Proper fractions
B1.1. Positive proper fractions
B1.1.1. Positive integers with integer exponents less than zero
B1.2. Negative proper fractions
B1.2.1. Negative integers with integer exponents less than zero
B2. Improper fractions (including mixed numbers)
B2.1. Positive improper fractions
B2.2. Negative improper fractions
C. Irrational numbers
C1. Transcendental numbers (cannot be defined as the root of a fraction or integer)
C2. Integer root of a positive integer for integers>1
C3. Integer root of a positive fraction for integers>1
C4. Fractional root of a positive integer
C5. Fractional root of a positive fraction
These are arbitrary sets that can be represented visually as circles. Some circles may be completely isolated from other circles; some may completely contain other circles; some may intersect other circles. Indentation above implies that the circle associated with the lesser indentation contains the whole of the circle with the greater indentation. For example, B1 is completely contained by B, B2.2 is contained in B2, which in turn is contained in B.
The separate sets of odd and even numbers could be included. We could add the set of prime numbers, positive integers>1 with no other factors than 1 and the number itself. We could add the set of perfect numbers, integers>1 in which the factors, including 1 but excluding the number itself, add up to the number. We could add factorials (the product of consecutive integers up to the factorial integer itself). These would be included in the superset of positive integers. The visual representation is that the subset is totally enclosed by the superset as a circle inside another circle.
An example of interlocking circles is the set of factorials with the set of perfect numbers, where 6 is contained in the overlap. 6 belongs to both sets.