- #1

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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?

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- Thread starter Ant farm
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- #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

HallsofIvy

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- #3

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[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

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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

JasonRox

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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

JasonRox

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I just realized that's what HallsOfIvy mentionned.

- #7

HallsofIvy

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That's ok. Thanks for the confirmation!

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