Knowlpedia

Basic properties of closure

Monotonicity, idempotence, and compatibility with finite unions

Proposition. Let (X,d)(X,d) be a metric space and let A,B⊂XA,B\subset X. Then:

  1. If A⊂BA\subset B, then A‾⊂B‾\overline{A}\subset \overline{B}.
  2. A‾=A\overline{A}=A if and only if AA is .
  3. A‾‾=A‾\overline{\overline{A}}=\overline{A} (idempotence).
  4. A∪B‾=A‾∪B‾\overline{A\cup B}=\overline{A}\cup \overline{B}.
Remarks

Proof sketch.

  1. Any closed set containing BB also contains AA, so the intersection defining A‾\overline{A} is contained in the intersection defining B‾\overline{B}.
  2. By definition A‾\overline{A} is closed and contains AA; equality holds exactly when AA is closed.
  3. A‾\overline{A} is closed, so closing it again does nothing.
  4. The inclusion A‾∪B‾⊂A∪B‾\overline{A}\cup\overline{B}\subset \overline{A\cup B} follows from monotonicity. For the reverse inclusion, note A‾∪B‾\overline{A}\cup\overline{B} is closed and contains A∪BA\cup B, hence contains A∪B‾\overline{A\cup B} by minimality.