# Determining if certain sets are open or closed.

I would like someone to verify my answers to the below questions are correct.

Exercise 3.2.8 from Understanding Analysis by Stephen Abbott.

Assume that $$A$$ is an open set, and $$B$$ is a closed set. Determine if the following sets are definitely open, definitely closed, both, or neither.

(a) $$\overline{A \cup B}$$

(b) $$A-B = \{x \in A: x \notin B\}$$

(c) $$(A^C \cup B)^C$$

(d) $$(A \cap B) \cup (A^C \cap B)$$

(e) $$(\overline{A})^C \cap \overline{A^C}$$

My Attempt.

(a) $$\overline{A}$$ is definitely closed for any set $$A$$. So, $$\overline{A \cup B}$$ is definitely closed.

(b) $$A - B$$ is definitely open. For all $$x$$ remaining in $$A - B$$, there exists an $$\epsilon$$-neighbourhood, that is contained entirely in $$A - B$$.

Consider $$A = (0,1)$$, $$B=\{1/n:n \in \mathbf{N}\} \cup \{0\}$$. $$A-B$$ is still open, because for the difference set is \begin{align*} \bigcup_{n=1}^{\infty}\left(\frac{1}{n},\frac{1}{n+1}\right) \end{align*} which is a countable union of open sets.

(c) $$(A^C \cup B)^C$$ is definitely open. $$A^C$$ is closed, so $$A^C \cup B$$ is closed and therefore $$(A^C \cup B)^C$$ is open.

(d) $$(A \cap B) \cup (A^C \cap B) = (A \cup A^C) \cap B = \mathbb{R} \cap B = B$$. This set is definitely closed.

(e) $$(\overline{A})^C \cap \overline{A^C}$$. I can't tell this one for sure, because its the intersection of an open set with a closed set.

• I think your b) needs more explanation (note $A\setminus B=A\cap B^C$). – David Mitra Jan 7 at 10:08
• @DavidMitra, can I argue: $A$ is open, $B^C$ is open, so finite intersection is open? – Quasar Jan 7 at 10:10
• Yes.${}{}{}{}{}$ – David Mitra Jan 7 at 10:11

For e): Since $$A$$ is open its complement is closed. So $$(\overline {A^{c})}=A^{c}$$. Now $$(\overline A)^{c} \cap A^{c}=(\overline A)^{c}$$ which is open.
• How is $(\overline{A})^C \cap A^C = (\overline{A})^C$? Perhaps, I'm missing something here. – Quasar Jan 7 at 10:12
• @Quasar Since $A$ is subset of its closure we see that $(\overline A)^{c}$ is a subset of $A^{c}$. So when you intersect these two we get $(\overline A)^{c}$ – Kavi Rama Murthy Jan 7 at 10:13