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

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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 0 ). This means that ( X, T ) is disconnected ( X, T 0 ) is disconnected or, equivalently, ( X, T 0 ) is connected ( X, T ) is disconnected when T 0 ⊃ 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 S A ( n ) = S A n is connected by [1, Thm 23.3] again. Ex. 23.3. Let A S 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 S 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 -
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This note was uploaded on 01/12/2011 for the course MATH 110 taught by Professor Brown during the Fall '08 term at Arizona Western College.

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

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