TITLE: What is the interior of a singleton?
QUESTION [8 upvotes]: I was wondering what can be said about the interior of $\{{4}\}$, the empty set?
The interior of a set $A$ is the largest open set contained by $A$. Hence, if the set at hand is a singleton, then isn't the interior of the singleton the empty set?

REPLY [2 votes]: It depends on what you mean by "open." Since $\lbrace 4\rbrace $ is open with respect to the topology on $\lbrace 4\ \rbrace$, $\text{int}(\lbrace 4 \rbrace)=\lbrace 4\rbrace$. If, for example, you mean open in $\mathbb{R}$, then $\lbrace 4\rbrace=[4,4]$, which is closed and not open, so $\text{int}(\lbrace 4\rbrace)=\emptyset$.