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From: macrakis@harvard.ARPA (Stavros Macrakis)
Newsgroups: net.math
Subject: (0,1) -> [0,1] bijection
Message-ID: <140@harvard.ARPA>
Date: Thu, 23-May-85 14:21:20 EDT
Article-I.D.: harvard.140
Posted: Thu May 23 14:21:20 1985
Date-Received: Sat, 25-May-85 05:33:32 EDT
References: <7368@ucbvax.ARPA> <376@h-sc1.UUCP>
Reply-To: macrakis@harvard.UUCP (Stavros macrakis)
Organization: Aiken Comp. Lab., Harvard
Lines: 10
Keywords: mapping

Problem: map (0,1) to [0,1] bijectively.

Insight: it's easy to map any countable set bijectively into any of
  its infinite subsets.

Solution: choose some ordering of ANY countable subset of (0,1).  Map
  the first and second elements to 0 and 1.  Map the nth element to the
  n-2nd element.  Leave everything else alone.

The earlier posted solution is a special case of this.