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

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- October 15th 2009, 12:15 PMamoebaUnion of Connected Sets
How do I use proof by contradiction to show that the union of two connected sets is connected?

- October 15th 2009, 12:32 PMredsoxfan325
- October 15th 2009, 12:35 PMLaurent
Note: you need to assume (for instance) that the two connected subsets have a non-empty intersection.

Let be connected sets such that . Assume by contradiction that is disconnected. Then there would exist two disjoint non-empty open subsets of such that (nb: are open for the induced topology on ). Let , . Then are disjoint open subsets of such that . Similarly, we can define for . Since and are connected, we must have either or , and either or . Each case leads to a contradiction. For instance, if and , then , whereas . If and , then and , hence ( are disjoint), which contradicts since . - October 15th 2009, 12:55 PMtonio
- October 15th 2009, 12:58 PMamoeba
- October 16th 2009, 04:04 PMLaurent
Because , , and are disjoint.

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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?

- October 16th 2009, 04:46 PMamoeba
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? - October 17th 2009, 01:28 AMLaurent
I did not. In fact, , as results directly from the definitions of and .

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