yifli
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Is my claim correct?
Boundary of any set in a topological space is compact
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Hi yifli! 
No, this is not correct, not even in the nice space [itex]\mathbb{R}[/itex]. Indeed, the set of rationals [itex]\mathbb{Q}[/itex] has a boundary which is entire [itex]\mathbb{R}[/itex] 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).
No, this is not correct, not even in the nice space [itex]\mathbb{R}[/itex]. Indeed, the set of rationals [itex]\mathbb{Q}[/itex] has a boundary which is entire [itex]\mathbb{R}[/itex] 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).