Boundary of Compact Closed Set is Compact
Jump to navigation
Jump to search
Theorem
Let $X$ be a topological space.
Let $K\subset X$ be a compact subspace of $X$.
Let $K$ be closed in $X$.
Then its boundary $\partial K$ is compact.
Proof
By Boundary of Set is Closed, $\partial K$ is closed in $X$.
By Set is Closed iff it Contains its Boundary, $\partial K \subset K$.
By Closed Set in Topological Subspace, $\partial K$ is closed in $K$.
By Closed Subspace of Compact Space is Compact, $\partial K$ is compact.
$\blacksquare$