# Math Help - Conectedness problem

1. ## Conectedness problem

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

Thank you!!

2. Originally Posted by Inti
Let $X$ be a connected topological space and $A \subset X$ connected. Suppose that $A^c = M \cup N$ with $M$ and $N$ separated (i.e. $cl(M) \cap N = \emptyset$ and $cl(N) \cap M = \emptyset$). Prove that $A \cup M$ and $A \cup N$ are connected.
First note that $A \cap M = \emptyset \;\& \;A \cap N = \emptyset$. WHY?
And $A \cup M \cup N = X$ and $X$ is connected.
If either $A \cup M$ or $A \cup N$ were not connected what about $X$?