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How do I show that the set A $\cup$ B where A = $\{(x,0) : 0 \le x \lt 1\}$ and B = $\{(0,y) : 0 \le y \lt 1\}$ is not a smooth manifold?

yDer
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    I assume you mean "with the differential structure inherited from the standard one on $\Bbb R^2$". Right? – Arthur Sep 15 '17 at 09:13

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If you look at him by itself he can as a matter of fact be given structure os smooth manifold, just by constructing a bijection with the real line and taking the differential structure of the real line. But if what you are asking is if this can be a submanifold of the plane, the the answer is NO, beacause when you describe the inclusion map of A ∪ B in the plane this is no an immersion (the differential at (o;o) has rank zero), therefore is not a regualar submanifold, not even an immerse submanifold.

kvicente
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