This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ....
View
Full
Document
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
 Brown
 Topology

Click to edit the document details