What's the closure of $(a,b)$ in discrete topology on the real number $\mathbb{R}$ In my opinion, by definition, the closure of $(a,b)$ in discrete topology on the real number $\mathbb{R}$ is $(a,b)$. However, I just saw the answer for this question is $[a,b]$. Now I am not sure which one.

A different approach is the following. The discrete topology is induced by the discrete metric.
Now sequences $x_n$ converge to $x$ in the discrete metric if and only if there exists $n_0\in \mathbb{N}$ such that $x_n=x$ for all $n\geq n_0$. Consequently, $(a,b)$ is the closure of itself in this topology.
