You have an interesting and perhaps useful conceptual point, but your language is at odds with current usage.<p>The concepts of infinite sizes of sets and point "at infinity" are different, and should be separated.<p>The thing is, there are multiple infinities even in the sense you're thinking about.  For example, you can have one at the end of every line in the complex plane.  That gives you the projective plane.  Again, in topology.<p>The one you're thinking of is the one-point compactification of the plane, but it's not the only one.