Is this set connected ?

[Shoshana Blum]

Let S1 be the set of points (x,y) in the plane R^2 such that both
x and y are rational numbers and let S2 be the complement of S1 in
R^2.

Is the set S2 connected ?

Thanks

[Fred Galvin]

Yes. Note that S1 is countable. The complement of any countable subset
of the plane is connected.

[Walter Hofmann]

Yes, it is even path-connected (as can be easily seen).

Walter

[Shoshana Blum]

>Yes. Note that S1 is countable. The complement of any countable
>subset of the plane is connected.

Hello thanks for the answer.
How can you prove the above that the complement of any countable
subset of the plane is connected ?

Shoshana

[Eckertson,Fred]

1) for all s in S, there is a line L, s in L and L is disjoint from S1.
Actually there are uncountably many such lines.

2) if s and t are in S there are non-parallel lines L_s and L_t
satisfying the conclusion of (1)

[Dave L. Renfro]

Walter Hofmann <spamd...@myrealbox.com>
[sci.math Mon, 16 Apr 2001 00:54:12 +0200]
<http://forum.swarthmore.edu/epigone/sci.math/blexspehwhin>

wrote

In fact, it's even polygonally connected.

Here's a hint for Shoshana Blum:

Given x,y in R^2 - Q^2, observe there are infinitely many lines
through x that miss Q^2 and there are infinitely many lines through
y that miss Q^2. {If you have to cheat, see Example 27.3(c) on
p. 198 of Willard [6].}

For others reading this who might be interested, this result was
first proved near the end of an 1882 paper by Georg Cantor [1].
See also the top of p. 48 of Cantor [2], pp. 85-86 of Dauben [3],
the translated letter from Cantor to Dedekind dated April 7, 1882
on pp. 871-872 of Ewald [4], and p. 196 of Ferreiros [5].

Ferreiros's remarks at the bottom of p. 196 are especially
interesting --->>>

********************************************************************

A few years later, Cantor published a new contribution that again
suggests the influence of Riemann's geometrical thought in his
mathematical work. The third installment of [Cantor 1879/84]
employs the non-denumerability of the real numbers to show that
there are discontinuous spaces in which continuous displacement
is possible. We have seen (%II.1.2) that Riemann approached the
question of physical space by means of a series of more and more
restrictive hypotheses. In a similar vein, Cantor had thought that
the continuity of physical space, which is not a necessity in
itself, is a consequence of continuous motion [Cantor & Dedekind
1937, 52]. But this conviction vanished once Cantor noticed that
in a space A, which is the result of subtracting a denumerable
dense set (e.g., the set of all point with algebraic coordinates)
from R^n, continuous displacement is still possible. This led him
to speculate about the possibility of a modified mechanics, valid
for such spaces A. This kind of physical application of his
mathematical speculations is reminiscent of Riemann's work on
geometry. Interestingly, that result of 1882 found application
years later in function theory, but Cantor never considered this
kind of use. {FOOTNOTE: Borel used it for a question of prolongation
of analytical functions in his Ph.D. thesis, which also reformulated
the notion of measure and gave the Heine-Borel theorem (see
[Hawkins 1980]).}

********************************************************************

[1] Georg Cantor, "Ueber unendliche, lineare Punktmannigfaltigkeiten"
(part 3), Math. Annalen 19 (1882), 113-121.

[2] Georg Cantor, "Contributions to the Founding of the Theory of
Transfinite Numbers", translated with additional notes by
Philip E. B. Jourdain, Dover Publications. [Reprint of original
1915 edition.]

[3] Joseph W. Dauben, "Georg Cantor: His Mathematics and Philosophy
of the Infinite", Princeton University Press, 1979.

[4] William B. Ewald, "From Kant to Hilbert: A Source Book in the
Foundations of Mathematics, vol. II, Clarendon Press, 1996.

[5] Jose Ferreiros, "Labyrinth of Thought: A History of Set Theory
and Its Role in Modern Mathematics", Science Networks / Historical
Studies 23, Birkhauser Verlag, 1999.

[6] Stephen Willard, "General Topology", Addison-Wesley, 1970.

Dave L. Renfro