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yifli
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Is my claim correct?
Boundary of any set in a topological space is compact
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In summary, a boundary in a topological space is the set of points that are on the edge or boundary between a set and its complement. Compactness in a topological space is defined as having a finite number of open sets that cover a set, rather than an infinite number. The boundary of any set in a topological space is always a closed set and therefore always compact. It is possible for a set in a topological space to have an empty boundary, which occurs when it is completely inside or outside of its complement. The compactness of a set is not directly related to its boundary, but the boundary can provide information about its compactness as a set with a non-empty boundary is always closed and therefore always 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).
FAQ: Boundary of any set in a topological space is compact
What is the definition of a boundary in a topological space?
The boundary of a set in a topological space is the set of points that are neither entirely inside the set nor entirely outside the set. In other words, it is the set of points that are on the edge or boundary between the set and its complement.
How is compactness defined in a topological space?
In a topological space, a set is considered compact if every open cover (a collection of open sets that covers the set) has a finite subcover (a subset of the open cover that still covers the set). This means that the set has a finite number of open sets that cover it, rather than an infinite number.
What is the relationship between the boundary and compactness in a topological space?
The boundary of any set in a topological space is always a closed set, and a closed set is always compact in a topological space. Therefore, the boundary of any set in a topological space is always compact.
Can a set in a topological space have an empty boundary?
Yes, it is possible for a set in a topological space to have an empty boundary. This would occur when the set is either completely inside or completely outside of its complement, and therefore there are no points on the edge or boundary between the two sets.
How is the compactness of a set related to its boundary in a topological space?
The compactness of a set is not directly related to its boundary in a topological space. A set can be compact without having a non-empty boundary, and a set can have a non-empty boundary without being compact. However, the boundary of a set can provide information about its compactness, as a set with a non-empty boundary is always closed and therefore always compact in a topological space.