Is the set of points in the plane whose coordinates are either both irrational, or both rational connected?
Connectedness of points with both rational or irrational coordinates in the plane?
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1What is the question? – 2012-04-23
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1@copper, I imagine the question is, consider the set of all points in the plane such that the coordinates are both rational, or both irrational. Is this set connected? But it would be nice if OP wrote it that way, instead of making us guess. – 2012-04-23
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0Sorry for ambiguous in the question. I edited it. – 2012-04-23
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0Can you mention the source/reference to the problem,it would be great! – 2017-10-22
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0@BAYMAX It has been a long time.. I forgot where I found this problem – 2017-10-22
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0Its ok,perhaps somebody else might be aware of it! – 2017-10-22
1 Answers
Let $S$ be your set, and suppose $S = U \cup V$ where $U$ and $V$ are disjoint, and are both closed and open in $S$. Note that if $x$ and $y$ are both rational, then the diagonal lines $\{(x+t, y+t): t \in {\mathbb R}\}$ and $\{(x+t,y-t): t \in {\mathbb R}\}$ are subsets of $S$. Using a path of diagonal line segments, it is possible to get from any point of ${\mathbb Q} \times {\mathbb Q}$ to any other while staying in $S$. Therefore one of $U$ and $V$, let's say $U$, contains all of ${\mathbb Q} \times {\mathbb Q}$. But ${\mathbb Q} \times {\mathbb Q}$ is dense in $S$, and $U$ is closed in $S$ so $U = S$.
For extra credit, show that $S$ is path-connected. In fact, if $a < b$ and $c < d$ with $(a,c)$ and $(c,d)$ in $S$, there is a continuous increasing function $f: [a,b] \to [c,d]$ such that $f(x)$ is rational if and only if $x$ is rational.
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0@MattN. This is not necessary (and one can argue it is better not to). – 2012-04-23
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1Can anyone prove the part about path-connectedness? Why does such a function $f(x)$ exist? – 2018-01-07
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2Take two-sided increasing sequences of rationals $x_n$ and $y_n$ with $x_n \to a$ and $y_n \to b$ as $n \to -\infty$, $x_n \to c$ and $y_n \to d$ as $n \to +\infty$, and join $(x_n, y_n)$ to $(x_{n+1},y_{n+1})$ with straight line segments. – 2018-01-08