I found your notation confusing and would not have understood without the diagram .I think you should say "We take our (metric) space to be the set of pairs $ (a,h(a))$ for all rational points $ a \in (0,1)^2$." And define $d$ accordingly. Then the meaning of $a_2$ is clear. And I believe you wish to show the fences don't intersect except at their edges. Can you do anything with reverse induction, on the number of crossings, or the number of jumps, in a counter-example?

I am accepting this for now without without studying as it will take me some time and I will have to review some topics in my copy of Kunen's Set Theory.Which gives me something to do for the next week. Thank you.

Very nice! In particular if a space $X$ ,e.g. $\beta N$, has a discrete subspace $Y$ with $|Y|=w(X)=2^{\omega}$ then $g(X)=g(D(2^{\omega}))$. It has also occurred to me to ask whether $g(X)$ can take singular values.Should I add this to my Q?

I got a lot of comments there but no progress so I decided to try the professional site.As I said on Math Exchange, last year by e-mail with Prof. Franklin Tall,he said he had not heard of this function.It just occurred to me to also ask whether g(T) can ever be singular.I am not a professional.

K. McAloon showed that if there exists an $\omega_2$ Aronszajn tree then $\omega_2$ is Mahlo in $L$.(Godel's constructible class.)

Prof.Franklin Tall ,a set theorist at U of T , said it was, presumably ,called that because it was believed that they are what Canadian set theorists are interested in.