A well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the well-order relation is then called a well-ordered set.
Now, We know that set $S = [0,1]$ is a Total-ordered set ( Assuming Relation defined on S is $ \leq $) as it is a Poset and every pair of elements are related(comparable) to each other. Now Since it is a Total-ordered set, All that we need to find whether it is a Well-ordered set or not, is that every non-empty subset of S must have a least element in this ordering.
But it is Not a Well-Ordered set as We can have at least one Subset of this Set which does not not any Least element. For example, What is the Least Element of $(0,1)$ ? There is no least element as whatever element you will say is least element, I can give you a number less than that in this set $(0,1)$. We can have more such subsets (like $(0.3,1)$) but having at least such subset of $S$ which doesn't have any least element is sufficient for it Not being Well-ordered set.
for real number there exist a subset (0,1) which is not well order because 0 is not present here isn't it ?
That is not the actual reason behind it not being Well-ordered set. The actual reason is that $(0,1)$ doesn't have any least element.
For integers it's well order set because we have 0 in it.
For integers $[0,1]$ is a Finite Set only with only Two elements $0,1$ and We have a theorem that "Every Finite Totally Ordered Set is Well-Ordered" (We can prove it easily but it's a simple theorem which can be visualized intuitively)
Again Having $0$ in it or not, is not the reason. The reason is that every non-empty subset of S must have a least element.
(I,<=) where I is set of integers is it a well order set and why?
No. It's Not Well-ordered set. Because Integers itself doesn't have any least element. What is the least element of Integers? or What is the least element of $(-∞, 0)$? ..Hence, $(I, \leq)$ is Not a Well-ordered set.
Note that Set of positive integers Or Set of Non-negative integers Or any set of integers $[n,∞)$ where $n$ is a specific(constant/fixed) integer value... All are Well-Ordered sets as every non-empty subset of these Sets has a least element.
