Instead of
removing middle thirds, imagine our construction as a process
of retaining the left and right thirds of each interval,
labeling our choices as we go along.
and the points
in the interval [0,1]. In order to do this, we'll take a fresh look
at our construction.
Instead of
removing middle thirds, imagine our construction as a process
of retaining the left and right thirds of each interval,
labeling our choices as we go along.
RRLR
RLRLRLRand so on...
What we arrive at is a strange "stairstep", with the steps labeled L and R.
Somewhere,
far below this diagram, lie the points of .
In order to reach a given point in ,
we would follow a "path" down this stairstep,
choosing left or right at each step, always careful to choose a subinterval
of our current "step". (You're not allowed to jump across a gap in the
current step; you just step down, choosing either the left or right
third of the current step.)
R
L
Land so on...
Any
given sequence of choices
necessarily determines a single point in
,
since the corresponding interval "steps" are
nested
and their lengths decrease to 0.
That is, the intersection of the
sequence of intervals in such a path will consist
of a single point in .
In other words, the left and right endpoints of our
sequence of "steps" will converge to a common value.
Conversely,
each point in
uniquely determines the path, or sequence of L's and R's,
that leads to it. Indeed, given a point in
,
at the "bottom" of our stairstep, there can be but one
sequence of interval "steps" that contain it. Why?
Because, at each horizontal level of our picture,
the L- and R-intervals are disjoint.
A given point in
can be in only one, unique subinterval at each level.
The
conclusion to be drawn from all of this is that we have
a matching, or correspondence between the points
of the Cantor set
and the collection of all sequences of L's and R's.
This
means that the Cantor set is HUGE. To see why, proceed to
even more on the Cantor set.
Or, quit now and be the only kid
on the block who doesn't know the secret!