SIMPLE QUESTION Open interval homeomorphic to R?

  • Thread starter fleazo
  • Start date
  • Tags
    Interval
In summary, the conversation discusses the difficulty of finding functions that are homeomorphisms between different types of intervals and sets. The suggestion is made to use translations, dilations, and tan to create a function from (a,b) to R. The idea is that size and scaling do not affect the topology of a space, making this a valid approach. The conversation also briefly mentions the graph of tan(x), but this does not seem to be relevant to the main topic of finding a function.
  • #1
fleazo
81
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
  • #2
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.
 
  • #3
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.
 
  • #4
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.
 
  • #5
SIMPLE ANSWER! Yes!
 
  • #6
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.
 
  • #7
have you looked at the graph of tan(x) lately?
 
  • #8
Not lately. I just checked and it looks like how I remember it. Am I missing something?
 

1. What does it mean for an open interval to be homeomorphic to R?

Homeomorphism is a mathematical concept that describes a continuous function between two topological spaces that has a continuous inverse. In the context of an open interval being homeomorphic to R (the set of real numbers), this means that there exists a continuous function between the two spaces that is one-to-one, onto, and has a continuous inverse.

2. How can I prove that an open interval is homeomorphic to R?

To prove that an open interval is homeomorphic to R, you can use the definition of homeomorphism and show that there exists a continuous function between the two spaces that satisfies the necessary conditions (one-to-one, onto, and has a continuous inverse).

3. What are the benefits of understanding the homeomorphism between an open interval and R?

Understanding the homeomorphism between an open interval and R can help in visualizing and understanding concepts in topology and real analysis. It also allows for the transfer of properties and theorems between the two spaces.

4. Can you give an example of a homeomorphism between an open interval and R?

One example of a homeomorphism between an open interval (0,1) and R is the function f(x) = tan((x-0.5)π). This function satisfies the necessary conditions of a homeomorphism and maps the open interval (0,1) to the set of real numbers.

5. Is there a difference between an open interval being homeomorphic to R and being isomorphic to R?

Yes, there is a difference. Isomorphism refers to preserving algebraic structures, while homeomorphism refers to preserving topological structures. In the context of an open interval and R, isomorphism would mean that the two spaces have the same algebraic properties, while homeomorphism means they have the same topological properties.

Similar threads

Replies
6
Views
2K
  • Topology and Analysis
Replies
12
Views
2K
Replies
45
Views
5K
  • Topology and Analysis
Replies
3
Views
1K
  • Topology and Analysis
Replies
7
Views
1K
Replies
2
Views
1K
  • Topology and Analysis
Replies
9
Views
2K
  • Topology and Analysis
2
Replies
54
Views
5K
  • Topology and Analysis
Replies
8
Views
1K
  • Topology and Analysis
Replies
9
Views
2K
Back
Top