TITLE: Why is the line at infinity one-dimensional?
QUESTION [13 upvotes]: Why is the line at infinity a one-dimensional manifold, i.e. why is it truly a "line" at infinity and not a plane? Is my reasoning below at all correct?
(When I say "dimension" I mean "real dimension" if talking about $\mathbb{RP}^2$ and "complex dimension" when talking about $\mathbb{CP}^2$ -- I do see why the complex line at infinity would be a two-real-dimensional manifold if the real line at infinity is a one-dimensional manifold.)
Naively, it seems like the line at infinity has two degrees of freedom, not just one. For either $\mathbb{RP}^2$ or $\mathbb{CP}^2$, it seems like one should have the following "basis" for the line at infinity: $$(1:0:0), (0:1:0) $$ 
Note: while writing up this question, I think I may have already thought up of an answer, so my attempt (which is long and not very rigorous) is included as a community wiki answer below, and I will add the (proof-verification) tag. 
However, I more or less only want to know any elegant answers/proofs which you may have, rather than whether or not my tentative  answer is correct.

REPLY [3 votes]: I think the key to unravelling your confusion may be in this line from the OP:

it seems like one should have the following "basis" for the line at
  infinity: $(1:0:0),(0:1:0)$

I think I get what you are saying:  any point on the line at infinity can be written in the form $(a:b:0)$, which seems to suggest that there are two independent coordinates.
What you are forgetting is that these are homogeneous coordinates, so that the point $(1:0:0)$ is the same point as $(2:0:0)$ and $(a:0:0)$ for any $a \ne 0$.  So $(1:0:0)$ does not point in a direction like a vector; any "multiple" of it is the exact same point.
More broadly, consider $(a:b:0)$.  Since we can't have both $a=0$ and $b=0$, at least one of them is nonzero.  Suppose that it is $b$.  Then $(a:b:0)$ can be written in the form $(r:1:0)$ for some $r \in \mathbb{R}$.  The set of all such points is in one-to-one correspondence with $\mathbb{R}$.
But what if $b=0$?  Then we know for sure that $a \ne 0$, and in this case we can write $(a:b:0) = (1:0:0)$, a single point.
So the line at infinity consists of two parts:

The set of points of the form $(r:1:0)$, which is in one-to-one correspondence with the elements of $\mathbb{R}$
The single point $(1:0:0)$, which is a "point at infinity" for the line at infinity.