$(X,d)$ metric space and $A\subset X$ and $A$ is connected. $$ \text{Card}(A) > 2 \implies \text{Card}(A) \geq \text{Card}(\mathbb{R}).$$
How do I prove it ?Waiting for your help?
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1If $R$ is $\Bbb R$, the set of real numbers, it’s false: $\Bbb R$ is a connected subset of itself. Do you mean that $\operatorname{card}A\ge\operatorname{card}\Bbb R$? – Brian M. Scott Jan 20 '15 at 18:36
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You can find a proof in my answer at http://math.stackexchange.com/questions/96541/connected-metric-spaces-with-at-least-2-points-are-uncountable – Mariano Suárez-Álvarez Jan 20 '15 at 18:46
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And here, here, and here. – Brian M. Scott Jan 20 '15 at 18:49