Proof of Equality for B=Int(B)∪bd(B) in Metric Space A

[gotjrgkr]

Let A be a metric space and B be a compact subset of A.
Is it true that an equality B = Int(B) \cup bd(B) holds??

 

[HallsofIvy]

In a metric space every compact set is closed (actually in every topological space). What does that tell you about its boundary points?

 

[gotjrgkr]

Boundary point is also limit point, right?
Then it belongs to B since B is closed?

 

[spamiam]

In a metric space every compact set is closed (actually in every topological space).

Hausdorff topological space, no? Which does include metric spaces, I admit.