Why is this line not homeomorphic to the unit circle?

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Discussion Overview

The discussion revolves around the concept of homeomorphism in topology, specifically examining why the interval ##[0, 2 \pi)## is not homeomorphic to the unit circle in ##\mathbb{R}^2##. Participants also explore the homeomorphism of knots, particularly the trefoil knot, in relation to the circle.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants argue that the interval ##[0, 2 \pi)## cannot be homeomorphic to the unit circle because the mapping does not have a continuous inverse, requiring the circle to be broken to map it back to the interval.
  • It is noted that homeomorphism requires topological continuity, which fails at the endpoints of the interval.
  • Participants mention that removing a point from the interval disconnects it, while the circle remains connected, and that the circle is compact whereas the interval is not.
  • One participant questions why the trefoil knot is considered homeomorphic to the circle, expressing confusion over the notion of deformation in relation to homeomorphism.
  • Another participant clarifies that homeomorphism is defined through continuous maps between topological spaces, not necessarily involving deformation, although deformation can serve as a heuristic.
  • There is a discussion about the standard of isotopy in knot theory, suggesting that under this criterion, knots and circles are not equivalent.

Areas of Agreement / Disagreement

Participants generally agree on the reasons why the interval and the circle are not homeomorphic, but there is disagreement regarding the interpretation of homeomorphism in the context of knots, particularly the trefoil knot.

Contextual Notes

Some limitations in the discussion include the dependence on definitions of homeomorphism and isotopy, as well as the unresolved nature of how deformation relates to homeomorphism in different contexts.

Mr Davis 97
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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.
 
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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.
 
Orodruin said:
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.
 
Homeomorphic requires topological continuity, which doesn't hold at the end points of the interval.
 
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.
 
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?
 
Mr Davis 97 said:
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.
 
Mr Davis 97 said:
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.
 
WWGD said:
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 ...
 
  • #10
Orodruin said:
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)
 
  • #11
WWGD said:
It would _knot_(??) be interesting? ;).
o:):H
 
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