Boundary of any set in a topological space is compact

[yifli]

Is my claim correct?

 

[micromass]

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).