Originally Posted by

**nano**
Something such as

, which was my main gripe, is undefined even on the extended reals.

Why is this a "gripe"? Did you not like my proof that

is undefined? Surely you recognize a proof by contradiction when you see one?

Anyway the extended reals were mentioned. Let's do this......

Suppose I take a segment

of the real line and join the two ends together. This we recognize as a circle; grand folk call it the 1-sphere

. Now since this is essentially the "same" as our line segment

, we might expect to be able to find a one-to-one correspondence between them. This correspondence - technically a "homeomorphism", since these are topological spaces - will be given by an invertible function from one to the other (this is called a bijection, btw).

Of course no such bijection exists; the map

sends the two ends of

to same point of

- our "join". And since well-behaved functions are not allowed to send a

*single* point in the domain to

*distinct* points in the co-domain, there can be no inverse

.

Now there is a theorem (due I think to Dedekind and Pierce) that states that an infinite set is always isomorphic to a proper subset of itself. Since

and since homeomorphism is an equivalence relation, by transitivity we might expect to be able to do something like the following.

Place our circle anywhere on the real line, and project each point of

"down" to a point in

in such a way that none of these projections intersect each other or

. (Let's recall that the standard topology on

mandates that the real line

is the open interval (or rather union of open intervals)

i.e. infinities are NOT included)

All goes swimmingly until we get to the North pole, where now our projections run parallel to

and seem never to meet it.

But we have the perfect construction specifically made for just this purpose, called the "projective real line". This is formed by adjoining what's called the "point at infinity" to

to form

. So now our point at the North pole projects to this point at infinity, and a homeomorphism

can be recovered, but at some cost; I cannot make this construction of 1-dimensional spaces without embedding them in, say a 2-dimensional space. And, if I am willing to do this, I have to say that parallel lines meet!

Worse still, since the two projections - "left" and "right" - from the North pole map to the same point in our projective line, this implies that, in

, the "end-points" of the projective real line

may be considered as the same point - that is

Which is no more, or less, than to assert that the projective real line has the same topology (up to homeomorphism) as the real circle.

And if we drop the topological constraints, we may say that

is the "extended reals". But note that whereas

is a field (both additive and multiplicative inverses exist), by the above,

is not a field, since there exists no

This line of thought can be extended to the plane

and its projective "partner"

, but here the geometry is a little more interesting; it's called "projective geometry", where "points" are "lines" and "lines" are "planes"

Gosh, I am a long-winded bastard. My excuse is that it is

*hammering* rain here in Northern France - this post is courtesy of McD's whose food looks like shite, but whose coffee is excellent