Is the Cantor function bijective from $[0,1]$ to Cantor set $K$? As $K$ is uncountable I think cardinality of $K$ must be $\mathfrak c$ as $K$ is a subset of $[0,1]$. But I am surprised whether there is any connection between cardinality and measure of a set. But in particular I am asking cardinality of $K$ and if it is $\mathfrak c$ give me a bijection with $K$ and $\mathbb R$.
2026-04-07 01:03:58.1775523838
Cardinality of cantor set $K$
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The Cantor function that I know is not a bijection. If $\frac 13 \lt x \lt \frac 23, f(x)=\frac 12$ It is true that the cardinality of $K$ is $\mathfrak c$. In the reals, all sets that have positive measure have cardinality $\mathfrak c$, but there are sets ($K$ among them) that have cardinality $\mathfrak c$ and measure $0$