Conectedness problem

**Inti** · Jul 14th 2009, 12:54 PM

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!!

**Plato** · Jul 14th 2009, 02:55 PM

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 $X$ ?

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