Closed sets and bounded sets
We cover each of the four possibilities below.
Closed and bounded: $[0,1]$
Closed and not bounded: $\cup_{n\in Z}[2n,2n+1]$
Bounded and not closed: $(0,1)$
Not closed and not bounded: $\cup_{n\in Z}(2n,2n+1)$
The integers as a subset of $\Bbb R$ are closed but not bounded.
$$\{x\in\mathbb R\mid x\geq 0\}$$
Also note that there are bounded sets which are not closed, for examples $\mathbb Q\cap[0,1]$.