I have to prove that every decomposable continuum contains a 2-od. These are some of the definitions:
A continuum is a nonempty compact, connected and metric space.
A continuum $X$ is said to be decomposable if there exist proper subcontinua, $A$ and $B$, of $X$ such that $X=A\cup B$.
A 2-od is a continuum $X$ such that there exists a subcontinuum, $C$, of $X$ with the property that $X\setminus C=E\cup F$, where $E$ and $F$ are nonempty mutually separated subsets of $X$ (in other words, $X\setminus C$ is disconnected).
This problem is the Excercise 14.19 form the book Hyperspaces of sets written by A. Illanes and S. B. Nadler.
I thought the following: Since $X$ is deconposable, we can write $X=A\cup B$ with $A$ and $B$ being proper subcontinua of $X$. Then, taking a point $p\in A\cap B$ (this point exists because $X$ is connected) I considered the component $K$ of $A\cap B$ that contains $p$, which is a subcontinua of $X$. Let $M$ be the union of the components of $A\cap B$ that are different from $K$. I thought that $T=(A\setminus M)\cup K \cup(B\setminus M)$ was a 2-od containd in $X$, but with some examples I saw that this is not true. Also in the book is proved that $C(X)$ is arcwise connected, and with this I know for example that I can find and order arc from $K$ to $A$ and also an order arc from $K$ to $B$ (these arcs being different), but i don't know if there is a way to continue with this idea.
Could you please give me some suggestion or some idea to solve the problem?