The formal definition of an interval

Question

I is A real interval iff ∀ x,y ∈ I the segment [x,y] ⊂ I

I can't understand why an interval is defined this way

Why it isn't defined the same way segments are? how can the definition of an object include the object itself in it? and how can I understand this definition ?

Answer 1

Following those definitions you will note that a segment is a special type of interval: it must be closed and bounded. Closed and bounded subsets of $\mathbb R^n$ are important in analysis because they provide compactness (don't worry about what this means, if you never heard of compactness).

Answer 2

Why [an Intervalle] isn't defined the same way Segments are?

The trick is that and interval $I$ can be "unlimited", like $[0,+∞)$, while a segment $[a,b]$ cannot.