In Chapter 1 of Pugh's Real Mathematical Analysis, Pugh gives the following picture:
I'm aware of other proofs to this like this one: bijection from (a,b) to R
but I'm interested in understanding how I should be interpreting this image. From top to bottom, i see the unit circle, the (-1, 1) line, the (a,b) line but I'm not sure what the lines connecting the intervals are meant to show. Would something like this be sufficient for a proof or is Pugh just trying to give intuition?
Best Answer
The diagram shows two bijections. The first, from $(a,b)$ onto $(-1,1)$, is a straightforward projection; call it $p$. Let $P$ be the point at the bottom where a bunch of lines converge. Then for each $x\in(a,b)$ we find $p(x)\in(-1,1)$ by drawing a line from $P$ through $x$ to the segment $(-1,1)$: the point where it hits the segment is $p(x)$.
The second bijection is from $(-1,1)$ to $\Bbb R$ and is a little more complicated; call it $g$. It uses the semicircle of radius $1$ centred at $C=\langle 0,1\rangle$ above the centre of the segment $(-1,1)$. If $x\in(-1,1)$, we find $g(x)$ as follows. First draw a vertical line up from $x$ until it meets the semicircle, say at a point $X$. Then draw a line from $C$ through $X$ to the straight line containing the segment $(-1,1)$; the point of intersection with that line is $g(x)$. As $x$ moves towards either end of the interval $(-1,1)$, the line from $C$ through $X$ gets closer and closer to horizontal, and $g(x)$ moves further off towards $-\infty$ or $+\infty$.
The function $f$ in the diagram is $g\circ p$, the composition of $p$ with $g$; it’s a bijection from $(a,b)$ to $\Bbb R$.