Proof of Quotient Map Q: A -> R for Exam Review

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Discussion Overview

The discussion revolves around the proof of the quotient map \( q: A \to \mathbb{R} \) derived from the projection map \( \pi_1: \mathbb{R} \times \mathbb{R} \to \mathbb{R} \) when restricted to a specific subspace \( A \). Participants explore the properties of this map, particularly its openness and closedness, in the context of an exam review.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant asserts that the set \( [0, \infty) \times \{0\} \) is both closed and open in \( A \), but its image under \( q \), which is \( [0, \infty) \), is neither closed nor open in \( \mathbb{R} \.
  • Another participant challenges this by stating that \( [0, \infty) \) is not closed and open because \( A \) is simply connected, suggesting that only \( A \) and the empty set are closed and open sets.
  • A participant proposes that to find examples of a closed set in \( A \) whose image under \( q \) is not closed, one might consider intersections of \( A \) with closed sets in \( \mathbb{R} \times \mathbb{R} \).
  • There is mention of a standard example involving the subset \( A = \{ (x,y) \in \mathbb{R}^2 : yx = 1 \} \) as a potential counterexample.
  • Another participant notes that quotient maps take saturated open (or closed) sets to open (or closed) sets, suggesting that considering subsets that are not saturated under the map may provide insight.

Areas of Agreement / Disagreement

Participants express differing views on the properties of the set \( [0, \infty) \) and the nature of closed and open sets in the context of the quotient map. The discussion remains unresolved regarding the specific examples needed to demonstrate the properties of the quotient map.

Contextual Notes

Participants highlight the need for examples that illustrate the behavior of closed and open sets under the quotient map, indicating that the discussion may depend on the definitions and properties of saturation in this context.

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Im reviewing material for the exam and came across this question:
Let pi_1:RxR->R be the projection on the first coordinate.
Let A be the subspace of RxR consisitng of all points (x,y) s.t either x>=0 or (inclusive or) y=0.

let q:A->R be obtained by resticting pi_1. show that q is quotient map that is neither open nor closed.
now to show that it's quotient map is the easy task, I want to see if I grasp it correctly, the set [0,infinity)x{0} is closed and open in A, but q([0,infinity)x{0})=[0,infinity) isn't closed nor open in R, correct?
 
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loop quantum gravity said:
the set [0,infinity)x{0} is closed and open in A, but q([0,infinity)x{0})=[0,infinity) isn't closed nor open in R, correct?


[0,inf)x{0} is certianly not closed and open because A is simply connected and so A itself and the empty set are the only closed&open sets.

I guess you will have to find two sets, one closed, the other open, such that their image is not closed (open).
 
I understand that [0,infinity) is closed in R, cause it's the complement of (-infinity,0) which is open in R.
So how to find such examples?
I mean for example a closed set in A would be an intersection of A with a closed set in RxR, now because q gives us only the first coordinate, then A obviously consists of all the points of the form: [0,infinity)xR and Rx{0}, which means that the map of such sets under q would be: subsets of [0,infinity) or R.
I don't see any example that shows what i need.
 
RxR, now because q gives us only the first coordinate, then A obviously consists of all the points of the form: [0,infinity)xR and Rx{0}, which means that the map of such sets under q would be: subsets of [0,infinity) or R.
I don't see any example that shows what i need.[/QUOTE].


I think the standard example is that of the subset A={ (x,y) in IR^2 : yx=1 }.




I don't know of a general way of generating (counter) examples, but there is

a result that quotient maps take saturated open ( equiv. closed) sets to

open (equiv. closed) sets. So if you can consider subsets that are not saturated

under your map, this should help.
 

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