Vanessa 's question at Yahoo Answers ( R^2-{(0,0)} homeomorphic to S^1 x R )

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In summary, the standard topology of $\mathbb R^2 - (0,0)$ is homeomorphic to $S^1 \times \mathbb R$. This is demonstrated by expressing $\mathbb R^2 \setminus \{(0,0)\}$ as a union of circles and using the fact that $(0,+\infty)$ is homeomorphic to $\mathbb R$. For more information, questions can be posted in the provided section.
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Hello Vanessa,

We can express $\mathbb R^2 \setminus \{(0,0)\}$ as disjoint union of circles: $$\mathbb R^2 \setminus \{(0,0)\}=\displaystyle\bigcup_{r\in (0,+\infty)}C_r\;,\qquad C_r=\{(x,y)\in\mathbb{R}^2:x^2+y^2=r^2\}$$ This is equivalent to say that $\mathbb R^2 \setminus \{(0,0)\}$ is homeomorphic to $S^1 \times (0,+\infty)$. Now, use that $(0,+\infty)$ is homeomorphic to $\mathbb{R}$.

If you have further questions, you can post them in the http://www.mathhelpboards.com/f13/ section.
 

FAQ: Vanessa 's question at Yahoo Answers ( R^2-{(0,0)} homeomorphic to S^1 x R )

What does the notation R^2-{(0,0)} mean?

The notation R^2-{(0,0)} refers to the set of all points in the two-dimensional Cartesian plane, except for the origin (0,0). In other words, it is the set of all points in the plane excluding the point at the intersection of the x-axis and y-axis.

What is meant by homeomorphic?

Homeomorphic is a mathematical term that describes a type of relationship between two objects or spaces. Specifically, it means that two objects or spaces have the same topological structure, meaning they can be transformed into each other without tearing, cutting, or gluing. In simpler terms, they have the same shape and structure, but may differ in size or orientation.

How is R^2-{(0,0)} homeomorphic to S^1 x R?

R^2-{(0,0)} and S^1 x R are homeomorphic because they have the same topological structure. This can be visualized by imagining the two-dimensional plane with a hole at the origin (R^2-{(0,0)}), and a cylinder (S^1 x R) with a circle at one end. By "stretching" or transforming the hole in the plane into a circle, and the cylinder into a flat plane, they become the same shape and structure.

What is the significance of R^2-{(0,0)} being homeomorphic to S^1 x R?

This homeomorphic relationship between R^2-{(0,0)} and S^1 x R has significant implications in mathematical and scientific fields such as topology, geometry, and physics. It allows for the application of concepts and theorems from one space to the other, making problem-solving and analysis easier and more efficient.

Is this relationship unique to R^2-{(0,0)} and S^1 x R, or are there other examples?

There are many other examples of spaces that are homeomorphic to each other, including the well-known example of a coffee mug and a donut (torus). The concept of homeomorphism is a fundamental concept in topology, and there are numerous other examples in mathematics and science where this relationship can be applied.

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