1. ## Closure (Metric Space)

Let A, B be subsets of a metric space. Show that $\overline{A \cup B} = \bar{A} \cup \bar{B}$ and that $\overline {A \cap B} \subset \bar{A} \cap \bar{B}$. Give an example to show that $\overline {A \cap B}$ and $\bar{A} \cap \bar{B}$ may not be equal.

2. Originally Posted by aliceinwonderland
Let A, B be subsets of a metric space. Show that $\overline{A \cup B} = \bar{A} \cup \bar{B}$ and that $\overline {A \cap B} \subset \bar{A} \cap \bar{B}$.
What are you having trouble with?
Give an example to show that $\overline {A \cap B}$ and $\bar{A} \cap \bar{B}$ may not be equal.
Consider when $A,B$ are seperated but share a limit point. Example $A=(-1,0),B=(0,1)$ because $A\cap B=\varnothing\implies\overline{A\cap B}=\varnothing$ but $0$ is a limit point of both so $\bar{A}\cap\bar{B}=\left\{0\right\}$

3. Originally Posted by aliceinwonderland
Let A, B be subsets of a metric space. Show that $\overline{A \cup B} = \overline{A} \cup \overline{B}$ and that $\overline {A \cap B} \subset \overline{A} \cap \overline{B}$. Give an example to show that $\overline{A \cap B}$ and $\overline{A} \cap \overline{B}$ may not be equal.

a) One inclusion: $\overline{A} \cup \overline{B}$ is closed since it is the finite union of closed sets and it contains $A \cup B$ so it must contain the closure $\overline {A \cup B}$.

Other inclusion: STS $\cup \overline{A_\alpha} \subseteq \overline{\cup A_\alpha}.$ Well, $\forall \alpha, A_\alpha \subseteq \overline{\cup A_\alpha}.$ So $\overline{A_\alpha} \subseteq \overline{\cup A_\alpha}$. Therefore $\cup \overline{A_\alpha} \subseteq \overline{\cup A_\alpha}.$

Next Part: $A \cap B$ is closed set containing $A \cap B$. So it must contain its closure $\overline{A \cap B}$

next part: take $A=(0,1)$ and $B=(-1,0)$ then $\overline{A \cap B} = \emptyset$ but $\bar{A} \cap \bar{B}=\{0\}$. QED