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Unformatted text preview: 11th December 2004 Munkres 25 Ex. 25.1. R ` is totally disconnected [Ex 23.7]; its components and path components [Thm 25.5] are points. The only continuous maps f : R R ` are the constant maps as continuous maps on connected spaces have connected images. Ex. 25.2. R in product topology: Let X be R in the product topology. Then X is is path con- nected (any product of path connected spaces is path connected [Ex 24.8]) and hence also connected. R in uniform topology: Let X be R in the uniform topology. Then X is not connected for X = B U where both B , the set of bounded sequences, and U , the complementary set of unbounded sequences, are open as any sequence within distance 1 2 of a bounded (unbounded) sequence is bounded (unbounded). We shall now determine the path components of X . Note first that for any sequence ( y n ) we have (0) and ( y n ) are in the same path component ( y n ) is a bounded sequence : Let u : [0 , 1] X be a path from (0) to ( y n ). Since u (0) = (0) is bounded, also u (1) = ( y n ) is bounded for the connected set u ([0 , 1]) can not intersect both subsets in a separation of 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.
- Fall '08