Exercise 1184Topology › Topology of ℝL1Added 3 Nov 2026

Interior, closure, boundary

#interior#closure#boundary

Statement

For $A=]0,1]\cup\{2\}$ , compute the interior, closure, and boundary of $A$ .

Hint

$\mathring A$ = union of open subsets of $A$ ; $\overline A$ = smallest closed set containing $A$ ; $\partial A=\overline A\setminus\mathring A$ .

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Interior. $\mathring A=]0,1[$ : $1$ is not interior (every ball around $1$ hits the complement), $2$ is not interior (isolated).

Closure. $\overline A=[0,1]\cup\{2\}$ : the limit points of $]0,1]$ are in $[0,1]$ ; $2$ is already in $A$ .

Boundary. $\partial A=\{0,1,2\}$ .

Conclusion. Interior $]0,1[$ , closure $[0,1]\cup\{2\}$ , boundary $\{0,1,2\}$ .