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# S23 - 27th January 2005 Munkres 23 Ex 23.1 Any separation X...

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27th January 2005 Munkres § 23 Ex. 23.1. Any separation X = U V of ( X, T ) is also a separation of ( X, T ). This means that ( X, T ) is disconnected ( X, T ) is disconnected or, equivalently, ( X, T ) is connected ( X, T ) is disconnected when T ⊃ T . Ex. 23.2. Using induction and [1, Thm 23.3] we see that A ( n ) = A 1 ∪ · · · ∪ A n is connected for all n 1. Since the spaces A ( n ) have a point in common, namely any point of A 1 , their union A ( n ) = A n is connected by [1, Thm 23.3] again. Ex. 23.3. Let A A α = C D be a separation. The connected space A is [Lemma 23.2] entirely contained in C or D , let’s say that A C . Similarly, for each α , the connected [1, Thm 23.3] space A A α is contained entirely in C or D . Sine it does have something in common with C , namely A , it is entirely contained in C . We conclude that A A α = C and D = , contradicting the assumption that C D is a separation Ex. 23.4 (Morten Poulsen). Suppose A X is open and closed. Since A is open it follows that X - A is finite. Since A is closed it follows that X - A open, hence

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S23 - 27th January 2005 Munkres 23 Ex 23.1 Any separation X...

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