Can say me why annulus and circle are not homeomorphic?

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In summary, an annulus is a shape formed by removing the interior of a circle, while a circle is a shape with a single continuous curve and no interior points. They are not homeomorphic because the annulus has a hole in the center while the circle does not. They are also not topologically equivalent because the annulus has one hole while the circle has zero holes. One way to visualize the difference is by imagining a rubber band stretched over a doughnut. In real life, examples of the annulus and circle not being homeomorphic include a bicycle wheel and a doughnut.
  • #1
seydunas
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Can say me why annulus and circle are not homeomorphic?
 
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Hi Seydunas, :smile:

The easiest way to see this is with so called "dispersion points". Removing two points from the circle gives you two connected components, thus it becomes disconnected. But from the annulus, you can remove any two points, and the result remains connected.

This means that the circle and the annulus are not homeomorphic.
 
  • #3


Owwwww, yes, i have tried to remove only one point but this did not give me any solution. Thank you micromass.
 

1. What is the difference between an annulus and a circle?

An annulus is a shape formed by removing the interior of a circle, leaving only the outer edge. A circle, on the other hand, is a shape with a single continuous curve and no interior points.

2. Why are the annulus and circle not homeomorphic?

Two shapes are considered homeomorphic if there is a continuous transformation that can be applied to one shape to make it exactly match the other. In the case of the annulus and circle, this is not possible because the annulus has a hole in the center while the circle does not.

3. Can the annulus and circle be topologically equivalent?

No, the annulus and circle are not topologically equivalent. Two shapes are topologically equivalent if they have the same number of holes and the same number of connected components. The annulus has one hole, while the circle has zero holes, making them topologically distinct.

4. How can we visualize the difference between the annulus and circle?

One way to visualize the difference is by imagining a rubber band stretched over a doughnut. The rubber band can be continuously deformed to resemble a circle, but it cannot be transformed to match the shape of the doughnut's hole. This illustrates the difference in topology between the annulus and circle.

5. Are there any real-life examples of the annulus and circle not being homeomorphic?

Yes, there are many real-life examples of the annulus and circle not being homeomorphic. For instance, a bicycle wheel and a doughnut are both circular in shape, but the bicycle wheel has a hole in the center while the doughnut has a hole in the middle. This shows that they are topologically different and not homeomorphic.

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