Apologies if this is an obvious question.
I am asking mostly to make sure that my assumptions are correct.
I find this notation in many Wikipedia articles without any definition, for example, see here.
I am assuming that it doesn't mean $x$ is $0$ or $1$.
Am I correct that it is refering to a mathematical interval.
To give context, I am trying to understand the Gautschi's Inequality where $s \in [0,1]$ which I am assuming means that $s$ can be any real number such that $0 < s < 1$
Yes, you are correct about it being an interval. So basically as the number of real numbers between 2 real numbers is infinite we denote the set of real number between the number a and b as an interval. The interval are of 3 types:
Open interval: It's a set of numbers between a and b, but the numbers a and b are excluded from the set. It is denoted by (a,b).
Closed interval: It's a set of numbers between a and b, including the numbers a and b. It is denoted as [a,b].
Half open and half closed: Here one of the number is included and other is not. Examples are [a,b) - here a is included but b isn't and in (a,b] b is included and a isn't.
In your question $x \in [0,1] \Rightarrow 0 \leq x \leq 1.$ And $x \in (0,1) \Rightarrow 0<x<1.$