- #1

- 19

- 0

Hey,

Just a tiny question, what is the space called:

Reals union infinity?

Just a tiny question, what is the space called:

Reals union infinity?

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- #1

- 19

- 0

Hey,

Just a tiny question, what is the space called:

Reals union infinity?

Just a tiny question, what is the space called:

Reals union infinity?

- #2

Science Advisor

Homework Helper

- 43,010

- 973

- #3

- 1,259

- 4

[tex] \mathbb{R} \cup \{\infty\} [/tex]

Which is called the real projective plane/line. You may also be looking for the affine extension of the real line:

[tex] \mathbb{R}\cup \{-\infty,\infty\} [/tex]

In either case we have that division by zero is allowed, but that the structure is no longer a field.

- #4

- 367

- 0

Equivalently,you can speak of the case where two parallel lines in euclidean plane intersects at infinitely distant point.Which is called the real projective plane/line. You may also be looking for the affine extension of the real line:

[tex] \mathbb{R}\cup \{-\infty,\infty\} [/tex]

Althought the introduction of an operative set of the points located in infinity

Last edited:

- #5

Homework Helper

Gold Member

- 2,372

- 4

As well as the space [itex] \mathbb{R}^2 \cup \{\infty\} [/itex] is another one-point compactification of R^2 so that we can create a homeomorphism from [itex] \mathbb{R}^2 \cup \{\infty\} [/itex] to the unit sphere in R^3 (or any other sphere). This well-known as stereographic projection where we call the sphere the Riemann's sphere. Probably familiar to anyone who has taken Complex Analysis.

I just read about this a few days ago, so I'm not 100% on how to say it properly, but that's the idea. I have yet to solve problems in the section and absorb the material more full. Lovely section in the book though.

Anyways, a homeomorphism is like an isomorphism (from Abstract Algebra) which preserves properties of the topological spaces. If you want to learn more about homeomorphisms, I strongly recommend just picking up any topology textbook. It's not too far in the textbook so you don't have to go too far. (It's in Chapter 2 of mine, and Chapter 1 was just about Set Theory.)

- #6

Homework Helper

Gold Member

- 2,372

- 4

I just realized that's what HallsOfIvy mentionned.

- #7

Science Advisor

Homework Helper

- 43,010

- 973

That's ok. Thanks for the confirmation!

Share:

- Replies
- 7

- Views
- 654

- Replies
- 7

- Views
- 656

- Replies
- 4

- Views
- 706

- Replies
- 19

- Views
- 93

- Replies
- 3

- Views
- 973

- Replies
- 5

- Views
- 2K

- Replies
- 11

- Views
- 965

- Replies
- 2

- Views
- 752

- Replies
- 20

- Views
- 4K

- Replies
- 2

- Views
- 5K