TITLE: Is $(a,a]=\{\emptyset\}$?
QUESTION [10 upvotes]: 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 $(a,a]=\{a \}$?

REPLY [4 votes]: 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 $(a,a]=\emptyset=\{\}$.