
Re: Dense subspaces of R^2
Posted:
Jun 23, 2009 9:30 AM


William Elliot wrote (in part):
> It is claimed that S = Q^2 \/ (R\Q)^2 is path connected. > How can the be?
I accidentally introduced the space S up in a May 2002 sci.math thread that you participated in, and I believe in one of your posts in that thread you came up with the limiting zigzag line segment method. ("Accidentally, because you began the thread asking about another dense subset of R^2, and I wasn't paying close enough attention to realize that the space you originally brought up was not the space my first post in that thread discussed.)
By the way, does anyone know of a reference to the pathwise connectedness of S (defined above) that predates Fall 1987? The earliest "reference" I know of is my own, when I proved that S is pathwise connected in a topology class (the actual problem assigned was to prove S is connected, but I managed to get pathwise connected).
For more about this, see the posts cited in the following August 2008 sci.math post of mine.
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Continuous real function with an "irrational" anomaly (25 August 2008) http://groups.google.com/group/sci.math/msg/39db777911fcbbc3
Back in Fall 1987 I used Cantor's result to prove that
Q^2 union P^2
is pathwise connected in R^2 (the plane), where Q is the set (subspace of R, actually) of rational numbers and P is the set of irrational numbers. For more about this, and how someone besides me wound up publishing it, see the first post below. Incidentally, I misread the problem being considered by the original poster in that thread, thinking it was the problem I had solved in 1987, but it was something different. However, I think the rest of the post is fine, except for one tangential comment I made in order to work in a Star Wars pun, which is taken care of in the 16 May 2002 post. The last post below includes an update where I found a simpler proof of the pathwise connectedness of Q^2 union P^2 in a recent topology text. [In all, you can find 3 proofs in these posts that this space is pathwise connected, and at least one proof independent of these 3 proofs that this space is connected.]
Topological Sandpaper [15 May 2002] http://groups.google.com/group/sci.math/msg/25850ce28ab37b19
Topological Sandpaper [16 May 2002] http://groups.google.com/group/sci.math/msg/4882cf8bae5f3352
R^2\Q^2 connected? [19 October 2005] http://groups.google.com/group/sci.math/msg/eb8c0d1057d06e73
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Dave L. Renfro

