The Cantor Set

In the next several pages we will examine a curious, but important example of an infinite set. The interest in this particular example is that it allows a variety of comparisons between the notions of "big" and "small". But more on this later. First, let's describe the set in question.

We begin with the closed interval [0,1].

Now we remove the open interval (1/3,2/3); leaving two closed intervals behind.

1/3

2/3

We repeat the procedure, removing the "open middle third" of each of these intervals, leaving us with four closed intervals.

1/9

2/9

1/3

2/3

7/9

8/9

And continue. At each stage we remove the middle thirds from the subintervals of the previous stage. It won't be possible to continue labeling points, so you'll have to use your imagination.

Although we can no longer draw the corresponding pictures (the intervals in our last picture have a width of only one pixel), this process is to be continued indefinitely.

The set we're interested in is the limit of this process. The set of points that remain after all these "middle thirds" have been deleted is called the Cantor set (named after Georg Cantor, 1845--1918). It's probably not at all clear that any points remain but, as we'll see, there are tons of points in the Cantor set!

In any event, this is our first piece of evidence concerning that "big versus small" thing... The Cantor set, whatever it is, must be awfully small; after all, we can't even draw it! But, if you trust me (and you really should), it apparently contains lots of points (I wouldn't lie to you!).

Proceed to more on the Cantor set, or simply return to my homepage.

Neal Carothers - carother@bgnet.bgsu.edu