In the nontrivial sense, does there exist a connected subspace of $\mathbb{R}^2$ which is a union of a non-empty countable collection of closed and pairwise disjoint line segments each of unit length, i.e. length $1$? What are some good examples, if any?
On the existence of a nontrivial connected subspace of $\mathbb{R}^2$
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general-topology
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0I was thinking of perhaps the set of fundamental parallelograms that tesselate the Cartesian plane, but that might not work unless we added the restriction that the sides of every parallelogram be of unit length. Hmmm… Actually it won't even work because the Cartesian plane itself is uncountable! – 2012-11-11
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1This might work: the union of $[0,1] \times 0$ with $a/b \times [a/b,a/b+1]$ for each $a/b \in \mathbb{Q} \cap [0,1]$, where $a/b$ is written in reduced form. – 2012-11-12
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0@Carl, that's a nice idea, but I don't think it will work: for some irrational $x$, the segments $x \times [x, x + 1]$ and $[x, x + 1] \times x$ will give us a disconnection. Also, I think you meant $\mathbb{Q} \cap (0, 1]$, because $0 \times [0, 1]$ meets $[0, 1] \times 0$ at the origin. – 2012-11-12
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1@HewWolff You're right. I meant for the line segments to be $a/b \times [1/b,1/b+1]$. I'm pretty sure it doesn't work as I wrote it there. – 2012-11-12