Why is this line not homeomorphic to the unit circle?

[Mr Davis 97]

I've been told that $[0, 2 \pi )$ is not homeomorphic to the unit circle in $\mathbb{R}^2$. Why not? From intuition, it would seem that I could just bend the line segment to fit the shape of a circle.

 

[Orodruin]

That map does not have a continuous inverse. More colloquially, you have to break the circle to map it back to your interval.

If you think it is "inuitive" that they should be homeomorphic then you unfortunately have created yourself a false intuitive image of what it means for two sets to be homeomorphic.

 

[Mr Davis 97]

That map does not have a continuous inverse. More colloquially, you have to break the circle to map it back to your interval.

If you think it is "inuitive" that they should be homeomorphic then you unfortunately have created yourself a false intuitive image of what it means for two sets to be homeomorphic.

Oh, I see. While I can form the segment into the circle, I cannot form the circle into the segment. And that inversion process from the circle to the segment has to be continuous also, when it is clearly not. I was just looking at the process in one direction.

 

[mathman]

Homeomorphic requires topological continuity, which doesn't hold at the end points of the interval.

 

[WWGD]

Removing one point from interval will disconnect it, while circle will remain connected ( and path connected) if you remove anyone point. In addition, circle is compact, while interval is not. Interval is contractible while circle is not. As others stated, by collapsing two points your map is not injective.

 

[Mr Davis 97]

One more question. Why is the trefoil knot homemorphic to the circle? In 3-space it does not seem that one can be deform a knot to a circle and vice versa. Is this notion of deformation how I should be thinking about homemorphism?

 

[Orodruin]

Is this notion of deformation how I should be thinking about homemorphism?

No. There is no reference to any sort of deformation in the definition of what a homeomorphism is - only continuous maps between topological spaces. It may be a good heuristic, but nothing then restricts you to three-dimensional space.

 

[WWGD]

One more question. Why is the trefoil knot homemorphic to the circle? In 3-space it does not seem that one can be deform a knot to a circle and vice versa. Is this notion of deformation how I should be thinking about homemorphism?

EDIT: By definition/construction, a knot is a homeomorphic image of the circle, and so each knot homeomorphic to it.The standard I am familiar with is that of (ambient) isotopy, a deformation through homeomorphisms, and not just through continuous functions. And, under that criteria, the two are not equivalent.

 

[Orodruin]

EDIT: By definition/construction, a knot is a homeomorphic image of the circle, and so each knot homeomorphic to it.The standard I am familiar with is that of isotopy, a deformation through homeomorphisms, and not just through continuous functions. And, under that criteria, the two are not equivalent.

Indeed, if knot theory was just about homeomorphisms, it would not be a very interesting field ...

 

[WWGD]

Indeed, if knot theory was just about homeomorphisms, it would not be a very interesting field ...

It would _knot_(??) be interesting? ;). ( sorry, I can't let a bad pun go to waste. Really, I can't)

 

[Orodruin]

It would _knot_(??) be interesting? ;).