The intersection of a countably infinite collection of half-open intervals of the same type may be empty, a half-open interval of that type, or a closed interval, which may be degenerate (a singleton) or not. Examples:
$\begin{align*} &\bigcap_{n\in\Bbb Z^+}\left(0,\frac1n\right]=\varnothing\\ &\bigcap_{n\in\Bbb Z^+}\left(-1,\frac1n\right]=(-1,0]\\ &\bigcap_{n\in\Bbb Z^+}\left(-\frac1n,\frac1n\right]=\{0\}\\ &\bigcap_{n\in\Bbb Z^+}\left(-\frac1n,1+\frac1n\right]=[0,1] \end{align*}$
If the intervals are $(a_n,b_n]$ for $n\in\Bbb Z^+$, let $a=\sup_na_n$ and $b=\inf_nb_n$; then
$\bigcap_{n\in\Bbb Z^+}(a_n,b_n]=\begin{cases} [a,b],&\text{if }a\notin\{a_n:n\in\Bbb Z^+\}\\\\ (a,b],&\text{if }a\in\{a_n:n\in\Bbb Z^+\}\;. \end{cases}$