Excluded point topological spaces

Question

The real numbers with $\tau=\{G:0\notin G\}\cup \{R\}$ be a topological spaces. What is the Int(Q) and Cl(Q)? I know the rational number Q is not open in $\tau$ since $0\in Q$.

Answer 1

The closure of $\Bbb Q$ is the smallest closed set containing it. $\Bbb Q$ is itself closed as its complement does not contain $0$. So the closure is just $\Bbb Q$.

The interior is the largest open subset and $\Bbb Q \setminus \{0\}$ is that set (it’s open by construction and the only larger subset is the set itself which is not open).

So in general $\operatorname{int}(A)$ is $A$ if $0 \notin A$ and $A\setminus \{0\}$ otherwise. Only $A=\Bbb R$ is an exception…