Is $(a,a]=\{\emptyset\}$?

Let $a \in \mathbb{R}$, and consider the half open interval $(a,a]$.

Is it correct to write this half open interval as $(a,a]=\{\emptyset \}$? Or $(a,a]=\{a \}$?

No, $(a,a]$ has no elements. It is empty. There is no real number $x$ such that $a < x \le a$. But you do not write that as $\{\varnothing\}$. You write it as $\varnothing$. See the difference?

No. Because $\emptyset$, the empty set, is not a real number, i.e. it is not an element of $\mathbb R$, the set $\{\emptyset\}$ is not a subset of $\mathbb R$. ON the other hand, $(a,a]$ is a subset of $\mathbb R$, so there can be no equality.

You are close, though, since $(a,a]$ is the empty set, so $(a,a]=\emptyset=\{\}$.