representation of the unit interval.
To see what remains after the process is completed:
Consider the
representation of the unit interval.
and
relabel Professor Carothers' diagram.
We next remove middle thirds from the four closed intervals, leaving eight closed intervals
Stated in terms of what we have removed, in the first stage, we remove all
points of the form
except
That
is with
in the first position after the decimal point, except
In the second stage we remove all points of the form
and
except
and
That is with
in the second position after the decimal point, except
In the third stage we remove all points of the form
,
, and
except
,
, and
That is with
in the third position after the decimal point, except
And so on....
In the end we are left with all points
whose
representation is decimal point followed by a string of
's
and
's,
or a finite string of
's
and
's
followed by a
followed by a string of
's
. Thus Cantor's Dust, as a set, is uncountable, it can be put in one to one
correspondence with the real numbers. (Think of the strings of
's
and
's
as strings of
's
and
's
, binary representations)
In terms of length:
In the first stage, we remove
open interval of length
.
In the second stage, we remove
interval's of length
, a total length of
.
In the
-th
stage , we remove
interval's of length
, a total length of
.
In all, we remove
Thus, in a way that can be made precise, Cantor's Dust has length
.
Homework, Due Tuesday January 31
Page 14 Problems 2.1, 2.3, and 2.7