Vol. 10 No 6 - Pi Mu Epsilon

476
PI Mu EPSILON JOURN
A VISUAL REPRESENTATION OF THE SEQUENCE SPACE 477
= ..
::I :&
u :.s :a : s
t& :1 = t:
....
u =
(0.15625, 1.0625) 2 (0.65625, 1.0625)
.. "
...
,. '
.. ·r-------____;.;_;;__~
"-:/
I
" . 16
=" / .. 1 '""'
(0.625,1) 32 (0.65625,1)
p 0 ,p 11 p 2 , ••• be a proper path from p to q. We will prove the case
re o and "'" differ in infinitely many terms, the finite case being similar.
Ld j ~ 1 and assume i is the index ofthejth term in which o and "'" differ.
We show that
I) x-l(pj) = totlta'"tl-ltls.J•ls.J•:z" ..
We proceed by induction on j.
Let j = 1 so i is the index of the first term where o and 1: differ. Since
path is proper, x- 1 (p 0
) and x- 1 (p 1
) must differ in the i th term, and in
other. Therefore, x- 1 (p 1 ) = s 0 s 1 ••• s 1 _ 1 t 1 s 1 • 1 ••• (since t 1 ~ s 1 ) and
further, since the i th term is the first term which they differ, sm = tm for
< i. Therefore x- 1 (p 1
) = t 0 t 1 • .. t 1 _ 1 t 1 s 1 • 1 s 1 • 2 .. ..
Now for the induction step. Let j > 1. Assume (1) is true. Let i be
the index of the j th term in which o and "'" differ. Let I be the index of the
(j + l)th term in which they differ. Then
x- 1 (pj) = t 0 t 1t 2 .•• t 1 _1tls.J. 1 s 1 • 2 ••• sx_ 1 SzSz•l''.
Since tz ~ Sz,
x- 1 (Pj. 1 ) = t 0t 1 t 2 ••• t.J_ 1t.Js 1 +1s 1 • 2 ••• sx_ 1 tzSz•l' ••
rJ :s
:: ! .1
"
(0.15625,0.0625)
:z ~=
:: ::
u 1&
u ~
u n
u :s
n ::
:: a
Figure 5 Proper Path from (0.15625,0.0625) to (0.625,1)
and since the I th term is the next term (after the i th term) in which they
differ, sm = tm fori < m < I. Therefore,
x-l (pj•l) = totlta. • • t.J-lt.Jt.J•lt.J+2' • • tx-ltzSz•lsi+'Z' • •
This completes the induction. Therefore the proper path is unique, as it is
the one prescribed by (1) . D
We can now use the lengths of the proper paths to induce a metric on
P.
The Metric
Definition 3.6. Define di> : P x P ~ R. by dP (p, q) = the length of the
unique proper path from p to q.
Theorem 3.4. (P, dP) is a metric space and the Crush Map,
... : (L, d"') ~ (P, dP), is an h:ometry.
Proof. By Lemma 3.1 x is a bijection, so it suffices to show that
d.., (o,"t) = dP(x(o) ,1C(1:)).
dP (x (a), 1t ("t)) = the distance of the proper path from 1t (o) to x ("t)
·.