Thread: Conectedness problem

Let $\displaystyle X$ be a connected topological space and $\displaystyle A \subset X$ connected. Suppose that $\displaystyle A^c = M \cup N $ with $\displaystyle M$ and $\displaystyle N$ separated (i.e. $\displaystyle cl(M) \cap N = \emptyset $ and $\displaystyle cl(N) \cap M = \emptyset $). Prove that $\displaystyle A \cup M$ and $\displaystyle A \cup N$ are connected.

Thank you!!

Originally Posted by Inti

Let $\displaystyle X$ be a connected topological space and $\displaystyle A \subset X$ connected. Suppose that $\displaystyle A^c = M \cup N $ with $\displaystyle M$ and $\displaystyle N$ separated (i.e. $\displaystyle cl(M) \cap N = \emptyset $ and $\displaystyle cl(N) \cap M = \emptyset $). Prove that $\displaystyle A \cup M$ and $\displaystyle A \cup N$ are connected.

First note that $\displaystyle A \cap M = \emptyset \;\& \;A \cap N = \emptyset $. WHY?
And $\displaystyle A \cup M \cup N = X$ and $\displaystyle X$ is connected.
If either $\displaystyle A \cup M$ or $\displaystyle A \cup N$ were not connected what about $\displaystyle X$?