Is an Open Interval Homeomorphic to R?

[fleazo]

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!

 

[quasar987]

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.

 

[phasic]

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.

 

[Bacle2]

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.

 

[Tinyboss]

SIMPLE ANSWER! Yes!

 

[phasic]

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.

 

[mathwonk]

have you looked at the graph of tan(x) lately?

 

[phasic]

Not lately. I just checked and it looks like how I remember it. Am I missing something?