
Topology Topology Math Forum 
 LinkBack  Thread Tools  Display Modes 
December 6th, 2008, 11:09 AM  #1 
Newbie Joined: Dec 2008 Posts: 4 Thanks: 0  discrete topology, product topology My friend and I are still stuck on: For each , let be the set , and let be the discrete topology on . For each of the following subsets of , say whether it is open or closed (or neither or both) in the product topology. (a) (b) (c) (d) (e) Recall that Progress: Take set (b). Let . If then there exists m such that f(m)=0. Then the set is an open neighbourhood of f contained in B. Therefore B is open. It's usually more difficult to check when a set is closed. You have to look at its complement and decide whether that is open. Sometimes this is straightforward. For example, the complement of set (a) is the set of all f such that f(10)=1. That is open, so set (a) is closed as well as open. For a slightly less easy example, look at set (c). Let . If then there exists m such that f(m)=f(m+1)=0. Then is an open neighbourhood of f containing no points of C. Therefore the complement of C is open and so C is closed. [color=red]I still do not know how to do parts (d.) and (e.)[/color] 

Tags 
discrete, product, topology 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Discrete sets and descrete topology  bigli  Topology  8  November 21st, 2013 11:54 AM 
Problem on product topology/standard topology on R^2.  vercammen  Topology  1  October 19th, 2012 12:06 PM 
Product Topology  matthematical  Topology  2  September 20th, 2011 03:20 PM 
discrete topology  toti  Topology  1  June 17th, 2010 02:58 PM 
discrete topology, product topology  Erdos32212  Topology  0  December 2nd, 2008 02:04 PM 