Is an Open Interval Homeomorphic to R?

  • Context: Graduate 
  • Thread starter Thread starter fleazo
  • Start date Start date
  • Tags Tags
    Interval
Click For Summary

Discussion Overview

The discussion centers on the topic of whether an open interval is homeomorphic to the real numbers, specifically exploring the existence of homeomorphisms between the interval (-1, 5) and R. Participants are engaged in mathematical reasoning and providing examples of potential homeomorphic functions.

Discussion Character

  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about finding a homeomorphism from the interval (-1, 5) to R and requests assistance.
  • Another participant suggests using a function that involves translation and dilation followed by the tangent function to establish a homeomorphism.
  • A different participant questions the validity of the previous suggestion but proposes a method involving the Cartesian plane and continuous bijections to map an interval to R.
  • One participant defends the previous suggestion, noting that rescaling does not affect the topology of a space.
  • Another participant critiques the initial suggestion, pointing out a potential error in the mapping process.
  • A participant prompts a discussion about the graph of the tangent function, indicating it may be relevant to the homeomorphism discussion.

Areas of Agreement / Disagreement

Participants express differing views on the validity of proposed homeomorphic functions, indicating that there is no consensus on the best approach or the correctness of the suggestions made.

Contextual Notes

Some participants' suggestions rely on specific transformations and mappings that may not be universally accepted or clearly defined, leading to potential misunderstandings about the functions proposed.

fleazo
Messages
80
Reaction score
0
Hi, I am having a major brain fart.

I realize that for example, open intervals and R are all topologically equivalent.


Similarly, closed, bounded intervals are topologically equivalent


And half open intervals and closed unbounded intervals are equivalent

But I am having a difficult time coming up with actual functions. For example, what is a function that would be a homeomorphism from (-1,5) --> R ?


I would REALLY appreciate some help here as my final is tomorrow morning!

Thanks!
 
Physics news on Phys.org
First take a function that send (a,b) to ((b-a)/2, (b-a)/2) simply by translation. Then take a dilatation that inflates of shrinks that to (-pi/2, pi/2). Then apply tan.
 
The above suggestion seemed a little off, but I did find a function from (a,b) > (-1,1) using the Cartesian plane using slope and evaluating for the 'intercept' at -1. Then I dilated by pi/2 and stretched with tan, mapping (a,b) onto R with a composition of continuous bijections.
 
I can't tell why you think the suggestion is off; seems pretty reasonable, since size/area/volume are not topological invariants , so that rescaling does not change
the topology of a space.
 
SIMPLE ANSWER! Yes!
 
First, he's sending the interval (a,b) to ((b-a)/2, (b-a)/2), which means sending (-2,1) to (-3,-3). If anything he's missing a negative sign.
 
have you looked at the graph of tan(x) lately?
 
Not lately. I just checked and it looks like how I remember it. Am I missing something?
 

Similar threads

Undergrad Collection of finite unions of half-open intervals form an algebra
  • · Replies 3 ·
Replies
3
Views
4K
High School Is the Quotient Topology of Real Numbers Homeomorphic to Real Numbers?
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Homemorphism in quotient topology
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Proving non homeomorphism between a closed interval & ##\mathbb{R}##
  • · Replies 45 ·
2
Replies
45
Views
7K
Undergrad An open set can be a closed interval?
  • · Replies 12 ·
Replies
12
Views
4K
MHB Open Sets in R .... .... Willard, Example 2.7 (a) .... ....
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Understanding the definition of continuous functions
  • · Replies 9 ·
Replies
9
Views
3K
Undergrad Heine-Borel Theorem shouldn't work for open intervals?
  • · Replies 9 ·
Replies
9
Views
3K
Undergrad Why Restrict Derivatives to Intervals?
  • · Replies 25 ·
Replies
25
Views
5K
Graduate Are open sets in R^n always homeomorphic to R^n?
  • · Replies 6 ·
Replies
6
Views
10K