Constraints on subset sum problem Subset sum is given by this question:
"The problem is this: given a set (or multiset) of integers, is there a non-empty subset whose sum is zero?"

My question is: If the numbers in the set are functions of other numbers, is that still subset sum? For example
The set {1,2,3} where the first number is X, the second is X+1, third is X+2 and so on. So it is a general set.

Is this allowed?
A set such as $\{1,2,3\}$ is valid input to the problem you defined, so yes, it is "allowed". When setting such constraints, you typically want to know "does the problem still remain NP-complete?". In this sense, notice there is a difference between "can my input take form such as this" and "if my input always takes this form, is the problem still NP-complete?" 

I believe you are after the latter; if you impose some constraints on the input, does the problem remain NP-complete? For example, if your input is the set $\{1,\ldots,n\}$ for some positive integer $n$, the answer is no, the problem can easily be decided in polynomial time (can you see why?)