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The set is given: $ \Theta = {(x, y) : (x, y) ∈ Q^c × Q^c}$ where $Q^c$ denotes the set of all irrational numbers. Is this set connected?

praeseo
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Galymbek
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1 Answers1

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No. The product $M_1 \times M_2$ is connected iff $M_1$ and $M_2$ are connected.

Lázaro Albuquerque
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  • @mathcounterexamples.net, the set of rationals (or its complement) is certainly not connected. – Paul Oct 28 '16 at 13:40