Cardinality between Open and Closed Sets

Date: 09/20/2001 at 00:01:16
From: Phillip Brubaker
Subject: Cardinality between open and closed sets

I would like to know how to prove that the sets (0,1) and [0,1] have 
the same cardinality. I know that they have the same cardinality as 
the set of all reals. I can prove that each set by itself has the same 
cardinality of all reals, and since a has same cardinality as x and b 
has same cardinality as x, a and b have same cardinality. But I would 
like to prove it another way without using the set of all reals.

Date: 09/20/2001 at 09:44:08
From: Doctor Rob
Subject: Re: Cardinality between open and closed sets

Thanks for writing to Ask Dr. Math, Phillip.

Set up a one-to-one correspondence f between the two sets as follows.
Let S = {1/(n+1): n in Z, n >= 1}, a countably infinite subset of 
(0,1). Define

   f(0) = 1/2,
   f(1) = 1/3,
   f(x) = 1/(2+1/x) = x/(2*x+1)   if x in S,
   f(x) = x                       if x in (0,1) but not in S.

Then f:[0,1] -> (0,1) is one-to-one and onto.

A minor variation on this shows that any finite number k of points
s[1], s[2], ..., s[k] can be added to an infinite set without
changing its cardinality. (Replace the first two equations above with 
f(s[i]) = 1/(1+i) for i = 1, 2, ..., k, and change 2 to k in the third 
equation above.) 

What this proves is that in cardinal number arithmetic, N + k = N for 
any infinite cardinal number N and any finite cardinal number k.

Actually, although I chose a specific S, any countably infinite subset 
of (0,1) can be used in a similar construction.

A variation on this shows that any countable number of points can be
added to an infinite set without changing its cardinality, or
N + aleph_0 = N for any infinite cardinal number N.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/