I'm clear with the definition of supremum and bounded sets. But for some reason, this statement in my lecture notes given by my Prof, doesnt seem to make sense.
Let $X = [0, 1) ∪ (2, 3]$. In this example the subset $[0, 1)$ is bounded above but it has no sup.
My question: Isn't 1 the supremum for $[0,1)$ ?
On
The issue here is when taking the $\sup$ you need to refer to the space you're in. Your space $X = [0, 1) \cup (2, 3]$ does not contain $1$ so $1$ cannot be the $\sup$ (in some sense claiming it's $1$ is as good as claiming it's elephant). It's bounded since $$ \forall x \in [0, 1) \; 0 \le x < 2.5 $$ and both $0, 2.5$ are in your space.
Does your professor maybe mean that $[0, 1)$ doesn't contain its supremum? It's axiomatic that every bounded subset of the reals has a supremum and infimum within the reals.
To specifically answer your question: yes, $\sup [0, 1) = 1$.