ANSWER and EXPLANATION
For a system of numbers to be closed for addition, subtraction, multiplication and division, it means that if each of those operations is carried out on two or more numbers from that system, then the result will ALWAYS be a member of that system.
So, let us answer for each system.
Natural numbers
Addition: Closed e.g. 5 + 3 = 8
Subtraction: Not closed e. g. 5 - 5 = 0 (not a natural number)
Multiplication: Closed
Division: Not closed e.g. 13 / 4 = 3.25 (not a natural number)
Integers
Addition: Closed
Subtraction: Closed
Multiplication: Closed
Division: Not closed e. g. 13 / -4 = -3.25 (not an integer)
Rational Numbers
Addition: Closed
Subtraction: Closed
Multiplication: Closed
Division: Not closed e. g. 3 / 0 = undefined
Irrational Numbers
Addition: Not closed
Subtraction: Not closed
Multiplication: Not closed
Division: Not closed
Real Numbers
Addition: Closed
Subtraction: Closed
Multiplication: Closed
Division: Not closed e.g. 3 / 0 = undefined