Definition of a circle in point set topology.

[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.