1. ## Disjoint connected

Suppose $\displaystyle A,B$ are connected subsets of some metric space $\displaystyle X$. Prove that if $\displaystyle A \cap B \neq \emptyset$ the $\displaystyle A \cup B$ is connected.

So suppose $\displaystyle A \cup B$ is not connected.

Then $\displaystyle A \cup B = C \cup D$ where $\displaystyle C$ and $\displaystyle D$ are disjoint and clopen. So $\displaystyle C = E_{O} \cap X$ and $\displaystyle C = E_{C} \cap X$. Also $\displaystyle D = F_{O} \cap X$ and $\displaystyle D = F_{C} \cap X$.

So $\displaystyle A \cup B = (E_{O} \cap X) \cup (F_{O} \cap X)$. Now what? How do you conclude that $\displaystyle A \cap B = \emptyset$?

2. Originally Posted by heathrowjohnny
Suppose $\displaystyle A,B$ are connected subsets of some metric space $\displaystyle X$. Prove that if $\displaystyle A \cap B \neq \emptyset$ the $\displaystyle A \cup B$ is connected.
So suppose $\displaystyle A \cup B$ is not connected.
Then $\displaystyle A \cup B = C \cup D$ where $\displaystyle C$ and $\displaystyle D$ are disjoint and clopen.
At that point, show that A must be a subset of either C or D.
Then B must be a subset of the other. There is your contradiction.

3. Originally Posted by Plato
At that point, show that A must be a subset of either C or D.
Then B must be a subset of the other. There is your contradiction.
Isn't this also contropostition? But this all depends on relative openess/closeness right?

Writing $\displaystyle C = E_{O} \cap X$ is what you should do? E.g. $\displaystyle E_O$ is an open set set. We have to show $\displaystyle C$ is open relative to $\displaystyle A \cup B$ right?

So want to show that $\displaystyle A \subseteq E_{O} \cap X$ and so on....?

4. Originally Posted by heathrowjohnny
We have to show $\displaystyle C$ is open relative to $\displaystyle A \cup B$ right?
I have no idea how your textbook/instructor requires you to prove connectivity.
The usual definition simply says that X is connected if and only if there do no exist two sets C & D such that $\displaystyle \begin{gathered} X \cap C \ne \emptyset \;,\;X \cap D \ne \emptyset \;\& \;X \subseteq C \cup D \hfill \\ \overline C \cap D = \emptyset \;\& \;\overline D \cap C = \emptyset \hfill \\ \end{gathered}$ .
Put simply: X is not a subset of the union of two sets, each having a nonempty intersection with X, and neither contains a point or a limit point of the other. That is known as a separation of X.
It seems that you are to use some other approach to connectivity.