S25 - 11th December 2004 Munkres 25 Ex. 25.1. R ` is...

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
This is the end of the preview. Sign up to access the rest of the 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.

Page1 / 2

S25 - 11th December 2004 Munkres 25 Ex. 25.1. R ` is...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online