Results 1 to 2 of 2

Math Help - Decomposition space - Quotient spaces

  1. #1
    Newbie
    Joined
    Nov 2010
    Posts
    12

    Decomposition space - Quotient spaces

    I want to prove that the decomposition space is the quotient topology.
    But I have a lot of trouble understanding the definition 9.5 (to be able to solve Theorem 9.6). I don't really understand what F and F are.

    Help appreciated!

    Here's the definition & the theorem:
    Decomposition space - Quotient spaces-scan-103530001.jpg
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by sssitex View Post
    I want to prove that the decomposition space is the quotient topology.
    But I have a lot of trouble understanding the definition 9.5 (to be able to solve Theorem 9.6). I don't really understand what F and F are.

    Help appreciated!

    Here's the definition & the theorem:
    Click image for larger version. 

Name:	Scan 103530001.jpg 
Views:	53 
Size:	279.3 KB 
ID:	20158
    Here's the idea. Take \mathbb{R} as a simple example. Then, we can form a decomposition as you call it (I'd call it a partition) by \mathcal{D}=\left\{(z,z+1):z\in\mathbb{Z}\}\cup\le  ft\{\{z\}:z\in\mathbb{Z}\right\} evidently \displaystyle \mathbb{R}=\coprod_{D\in\mathcal{D}}D. So, I want to topologize \mathcal{D} by saying that \mathcal{F}\subseteq\mathcal{D} is open if and only if \displaystyle \bigcup_{F\in\mathcal{F}}F is open in \mathbb{R}. For example, \left\{(z,z+1):z\in\mathbb{Z}\right\}=\mathcal{F} is open since \displaystyle \bigcup_{F\in\mathcal{F}}F=\bigcup_{z\in\mathbb{Z}  }(z,z+1)=\mathbb{R}-\mathbb{Z} which is open. Whereas \displaystyle \mathcal{F}'=\left\{\{z\}:z\in\mathbb{Z}\right\} is not open since \displaystyle \bigcup_{F\in\mathcal{F}'}F=\bigcup_{z\in\mathbb{Z  }}\{z\}=\mathbb{Z} is not open in \mathbb{R}. Make sense?


    For the actual exercise, if it makes it easier for you to think about it, define [tex]\sim [tex] on \mathbb{R} by saying x\sim y if and only if there exists D\in\mathcal{D} such that x,y\in D. Then, the quotient map is \pi:\mathbb{R}\to\mathcal{D}:x\mapsto[x] where [x] is evidently the equivalence class of x under \sim.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Quotient space
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: September 28th 2011, 01:20 PM
  2. [SOLVED] Quotient spaces
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: December 19th 2010, 12:28 PM
  3. Quotient spaces and equivalence relations
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: March 10th 2009, 01:01 AM
  4. homeomorphic quotient spaces
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 21st 2009, 07:51 AM
  5. quotient spaces
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: November 4th 2008, 09:45 AM

Search Tags


/mathhelpforum @mathhelpforum