1. ## Disjoint connected

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

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

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

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

2. Originally Posted by heathrowjohnny
Suppose $A,B$ are connected subsets of some metric space $X$. Prove that if $A \cap B \neq \emptyset$ the $A \cup B$ is connected.
So suppose $A \cup B$ is not connected.
Then $A \cup B = C \cup D$ where $C$ and $D$ 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 $C = E_{O} \cap X$ is what you should do? E.g. $E_O$ is an open set set. We have to show $C$ is open relative to $A \cup B$ right?

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

4. Originally Posted by heathrowjohnny
We have to show $C$ is open relative to $A \cup B$ 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 $\begin{gathered} X \cap C \ne \emptyset \;,\;X \cap D \ne \emptyset \;\& \;X \subseteq C \cup D \hfill \\ \overline C \cap D = \emptyset \;\& \;\overline D \cap C = \emptyset \hfill \\ \end{gathered}$ .
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.