# Thread: Union of Connected Sets

1. ## Union of Connected Sets

How do I use proof by contradiction to show that the union of two connected sets is connected?

2. Originally Posted by amoeba
How do I use proof by contradiction to show that the union of two connected sets is connected?
The statement as it is now is false.

Consider $\displaystyle A=[0,1]$ and $\displaystyle B=[2,3]$. $\displaystyle A$ and $\displaystyle B$ are clearly connected, but $\displaystyle A\cup B$ isn't.

You would have to assume that $\displaystyle A$ and $\displaystyle B$ are not disjoint, that is, $\displaystyle \exists~x$ such that $\displaystyle x\in A$ and $\displaystyle x\in B$.

3. Originally Posted by amoeba
How do I use proof by contradiction to show that the union of two connected sets is connected?
Note: you need to assume (for instance) that the two connected subsets have a non-empty intersection.

Let $\displaystyle X,Y$ be connected sets such that $\displaystyle X\cap Y\neq\emptyset$. Assume by contradiction that $\displaystyle X\cup Y$ is disconnected. Then there would exist two disjoint non-empty open subsets $\displaystyle O_1,O_2$ of $\displaystyle X\cup Y$ such that $\displaystyle O_1\cup O_2=X\cup Y$ (nb: $\displaystyle O_1,O_2$ are open for the induced topology on $\displaystyle X\cup Y$). Let $\displaystyle U_1=O_1\cap X$, $\displaystyle U_2=O_2\cap X$. Then $\displaystyle U_1,U_2$ are disjoint open subsets of $\displaystyle X$ such that $\displaystyle U_1\cup U_2=X$. Similarly, we can define $\displaystyle V_1,V_2$ for $\displaystyle Y$. Since $\displaystyle X$ and $\displaystyle Y$ are connected, we must have either $\displaystyle U_1=\emptyset$ or $\displaystyle U_2=\emptyset$, and either $\displaystyle V_1=\emptyset$ or $\displaystyle V_2=\emptyset$. Each case leads to a contradiction. For instance, if $\displaystyle U_1=\emptyset$ and $\displaystyle V_1=\emptyset$, then $\displaystyle U_1\cup V_1=\emptyset$, whereas $\displaystyle U_1\cup V_1=O_1\neq\emptyset$. If $\displaystyle U_1=\emptyset$ and $\displaystyle V_2=\emptyset$, then $\displaystyle X\cap O_1=\emptyset$ and $\displaystyle Y\cap O_2=\emptyset$, hence $\displaystyle (X\cap Y)\cap(O_1\cup O_2)=\emptyset$ ($\displaystyle O_1,O_2$ are disjoint), which contradicts $\displaystyle X\cap Y\neq \emptyset$ since $\displaystyle O_1\cup O_2=X\cup Y$.

4. Originally Posted by amoeba
How do I use proof by contradiction to show that the union of two connected sets is connected?

you cannot prove that because it isn't true: for example (0,1) \/ (1,2) is the union of two connected sets but it isn't connected

Tonio

5. Originally Posted by Laurent
Let $\displaystyle U_1=O_1\cap X$, $\displaystyle U_2=O_2\cap X$. Then $\displaystyle U_1,U_2$ are disjoint open subsets of $\displaystyle X$ such that $\displaystyle U_1\cup U_2=X$ .
How do you know that U1 and U2 are disjoint?

Also, you said

Then U_1,U_2 are disjoint open subsets of X such that U_1\cup U_2=X.

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How do you know this fact that the union of U1 and U2 is X?

6. Originally Posted by amoeba
How do you know that U1 and U2 are disjoint?
Because $\displaystyle U_1\subset O_1$, $\displaystyle U_2\subset O_2$, and $\displaystyle O_1,O_2$ are disjoint.

Also, you said

Then U_1,U_2 are disjoint open subsets of X such that U_1\cup U_2=X.

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How do you know this fact that the union of U1 and U2 is X?
Because $\displaystyle X\subset X\cup Y = O_1\cup O_2$. You should draw a diagram if you don't get it. I just split $\displaystyle X$ into two parts according to the partition of $\displaystyle X\cup Y$. If $\displaystyle O_1,O_2$ cut $\displaystyle X\cup Y$ in two disjoint pieces, then their intersections with $\displaystyle X$ cut it also in two disjoint pieces.

7. How come you assumed in your post that O1 U O2 = U1 U U2?

Why isn't showing that the pair S1, S2 isn't a disconnection sufficient?

8. Originally Posted by amoeba
How come you assumed in your post that O1 U O2 = U1 U U2?
I did not. In fact, $\displaystyle U_1\cup U_2= X$, as results directly from the definitions of $\displaystyle U_1$ and $\displaystyle U_2$.

Why isn't showing that the pair S1, S2 isn't a disconnection sufficient?
Which $\displaystyle S_1, S_2$ ?? Since you procede by contradiction, you must find a disconnection of either $\displaystyle X$ or $\displaystyle Y$, this would be the contradiction.

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# prove that the union of two connected sets having non empty intersection is connected

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