What is Real's Union Infinity?

[Ant farm]

Hey,

Just a tiny question, what is the space called:

Reals union infinity?

 

[HallsofIvy]

I'm not sure of your question. There exist several different ways it could be interpreted. If you are trying to extend the real numbers by adding "infinite numbers" (and "infinitesmals") then, depending on how you do it, you might have the "hyper-reals" or "ultra-reals". If you are not concerned with algebraic properties but just want to add a single point "at infinity" (which is "close" to extremely large numbers whether positive or negative) to make it geometrically "nicer" (technically compact) and is topologically equivalent to a circle, that is the "one point compactification" of the reals. If you want to add "positive infinity" and "negative infinity" at each end so it is topologically equivalent to a closed line segment, that is the "Stone-Chech compactification" of the reals.

 

[Crosson]

I think ant farm is quit literally referring to the space:

\mathbb{R} \cup \{\infty\}

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

\mathbb{R}\cup \{-\infty,\infty\}

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

 

[tehno]

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

\mathbb{R}\cup \{-\infty,\infty\}

Equivalently,you can speak of the case where two parallel lines in euclidean plane intersects at infinitely distant point.
Althought the introduction of an operative set of the points located in infinity per definition may have some strange properties (for instance. line in euclidean plane to have only one infinitely distant point-not two as one can think following extensions to -\infty and +\infty),the approach turns to be very useful in Projective Geometry and its' applications.

 

[JasonRox]

Another way to look at it is that \mathbb{R} \cup \{\infty\} is the one-point compactification of R so that we can create a homeomorphism from \mathbb{R} \cup \{\infty\} to the unit circle in R^2 (or any circle).

As well as the space \mathbb{R}^2 \cup \{\infty\} is another one-point compactification of R^2 so that we can create a homeomorphism from \mathbb{R}^2 \cup \{\infty\} 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.)

 

[JasonRox]

I just realized that's what HallsOfIvy mentionned.

 

[HallsofIvy]

That's ok. Thanks for the confirmation!