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Topic: Dense subspaces of R^2
Replies: 22   Last Post: Jun 26, 2009 5:25 AM

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Dave L. Renfro

Posts: 4,792
Registered: 12/3/04
Re: Dense subspaces of R^2
Posted: Jun 23, 2009 9:30 AM
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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
zig-zag 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.


Continuous real function with an "irrational" anomaly (25 August 2008)

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]

Topological Sandpaper [16 May 2002]

R^2\Q^2 connected? [19 October 2005]


Dave L. Renfro

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