Definition of a circle in point set topology.

[jgens]

There is no grief

Then why start a thread about avoiding metric structures on the sphere?!

and yes, in principle that is possible, by specifying the set's of it's topology.

Now you have my interest. Would you mind specifying these sets without referencing a metric on Rⁿ or the ordering on R?

 

[center o bass]

Then why start a thread about avoiding metric structures on the sphere?!

I did not! If you had looked back, which I asked you to earlier, you would see that the thread was started with a question on how one might define a circle in topology. TinyTim answered in terms which did not involve the unit circle in R^2 and I felt it a natural progression to ask how one could do the same for a circle as a manifold. This was the point. You came in by arresting me on a terminological error.

Now you have my interest. Would you mind specifying these sets without referencing a metric on Rⁿ or the ordering on R?

Yes here is the pair $(\mathbb{R}^n, \mathcal{T})$. There is often a difference between principle and practice.

 

[jgens]

I did not! If you had looked back, which I asked you to earlier, you would see that the thread was started with a question on how one might define a circle in topology.

I still contend that defining S¹ = R/Z or S¹ = I/∂I suffers from the same defects as the unit circle definition. But at this point in time I am more interested in your later claim...

Yes here is the pair $(\mathbb{R}^n, \mathcal{T})$. There is often a difference between principle and practice.

But what are the sets in T? You claimed you could tell me what sets belong in T without reference to a metric on Rⁿ or the ordering on R. I am still waiting for this.

 

[center o bass]

I still contend that defining S¹ = R/Z or S¹ = I/∂I suffers from the same defects as the unit circle definition. But at this point in time I am more interested in your later claim...

I'm not, because that was not the point of the thread.

But what are the sets in T? You claimed you could tell me what sets belong in T without reference to a metric on Rⁿ or the ordering on R. I am still waiting for this.

No. I claim that in principle we can topologize any space $X$ by providing a pair $(X, \mathcal{T})$. How one might construct $\mathcal{T}$ in practice is another question.

 

[jgens]

How one might construct $\mathcal{T}$ in practice is another question.

With this in mind hopefully you can understand why I contend that virtually any definition of S¹ is going to require some additional structure in its definition. You can hide or obscure that structure a small amount but it is still there. In any case, just a friendly FYI if you want a definition of S¹ with a minimal amount of structure, then the R/Z definition you have been championing is not the best example. That definition induces a group structure on the circle which is something not all manifolds have.

Edit: Basically I am saying that any definition of S¹ you choose is just as good as another from a point-set topological standpoint. They all have their uses in point-set topology for different problems. To single one definition out as problematic because it contains extra structure is silly because virtually any way of topologizing the circle gives you a bit more information than its topology alone.

 

[lavinia]

I guess I could rephrase my question as: Can one define the circle, as a manifold, as the unit interval with points identified? In that case what charts should one use to specify it's differential structure?

No. The unit interval with its points identified does not define a manifold because it does not provide coordinate charts. You need to show that it can be given the structure of a manifold

 

[lavinia]

BTW: The unit interval with end points identified is an example of a structure called a CW complex. A homeomorphism from this space to a circle shows that a circle is a CW complex.

 

[center o bass]

No. The unit interval with its points identified does not define a manifold because it does not provide coordinate charts. You need to show that it can be given the structure of a manifold

Alright. But you can provide charts for it, can't you? I.e. one can give it a differential structure. How would that be done?

 

[jgens]

Let U₁ = (0,1) and U₂ = [0,1/2)∪(1/2,1]. Let φ₁ = id and φ₂|[0,1/2) = id and φ₂|(1/2,1] = 1-id. Under the 0~1 identification these descend to a pair of charts on the circle.