This is exercise 6 on page 101 in Chapter 17 of James R. Munkers’ Topology 2nd Edition textbook.

Question

Let \(A, B,\) and \(\mathcal{A_\alpha}\) denote subsets of a space \(X\). Prove the following.

(a) If \(A \subset B\), then \(\overline{A} \subset \overline{B}\)

Let \(x \in \overline{A}\), there must be closed sets containing \(A\) but not containing all of \(B\). Thus, \(x \notin \overline{A}\) for all \(y \in B\). Finally, \(\overline{A} \subset B\).

Thus \(A \subset \overline{A} \subset B \subset \overline{B}\) and \(\overline{A} \subset \overline{B}\).

\[\tag*{$\blacksquare$}\]

(b) Prove \(\overline{A \cup B} = \overline{A} \cup \overline{B}\)

For \(\overline{A \cup B} \subset \overline{A} \cup \overline{B}\)

\(A \cup B \subset \overline{A \cup B}\). Also, \(A \subset \overline{A}\) and \(B \subset \overline{B}\). Thus, \(\overline{A \cup B} \subset \overline{A} \cup \overline{B}\).

\[\tag*{$\blacksquare$}\]

Extras

link to the mathjax LaTeX specification: https://treeofmath.github.io/tex-commands-in-mathjax/TeXSyntax.htm