general topology – Is the empty set compact in $mathbb R$?

Here they say that the emptyset is compact. Nevertheless, in $mathbb R$, I know that compact sets are closed and bounded. So, indeed $varnothing $ is closed, but we have that $$inf_{mathbb R}varnothing =+infty quad text{and}quad sup_{mathbb R}varnothing =-infty ,$$
in partuclar, it doesn’t seem bounded. So, who is correct ?