# Boundary of any set in a topological space is compact

## Main Question or Discussion Point

Is my claim correct?

Hi yifli!

No, this is not correct, not even in the nice space $\mathbb{R}$. Indeed, the set of rationals $\mathbb{Q}$ has a boundary which is entire $\mathbb{R}$ and is thus not compact!

The result is true in compact topological spaces, however. (because any closed set in a compact space is compact, and because the boundary is always a closed set).