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Is the cardinality of $\mathcal{C}[0,1]$ the same as the cardinality of $\mathbb{R}$?

My attempt: I know that $[0,1] ,(0,1) $ are uncountable as $\mathbb{R}$ are also uncountable so $[0,1]$ have same cardinality as $\mathbb{R}$

I am in confusion about $\mathcal{C}[0,1]$ where $\mathcal{C}[0,1]$ is the space of continious real value function on the interval $[0,1]$.

Any hints/solutions?

peterh
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jasmine
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1 Answers1

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Each continuous function on $[0,1]$ is determined by its values on the countable set $\Bbb Q\cap[0,1]$. Therefore $$|C[0,1]|\le|\Bbb R|^{\aleph_0}=(2^{\aleph_0})^{\aleph_0} =2^{\aleph_0\times\aleph_0}=2^{\aleph_0}=|\Bbb R|.$$

Angina Seng
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    Moreover, there's an obvious bijection between $\mathbb R$ and the constant functions, which are a subset of $C[0,1]$, so $|\mathbb R| \leq |C[0,1]|$. –  Sep 13 '18 at 17:32