Choice is not needed.


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EDIT: Thanks to @dfeuer for pointing out my original argument required Dependent Choice, and for helping me arrive at a choice-less proof.

EDIT2: I discovered a new flaw in the `proof', namely that $x^*$ could be an element of $C_n$ (this should to be avoided to guarantee empty intersection of$C_n$'s). Will work to fix.

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I will say $X$ is *countable* if there is an injection $f:X\to \omega$, where $\omega$ is the set of natural numbers. *Uncountable* means not countable.

By a *continuum* I shall mean a connected compact Hausdorff space.

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**Theorem (ZF).** Every non-degenerate continuum is uncountable.



***Proof***. Let $X$ be a non-degenerate continuum.

For a contradiction suppose $X$ is countable.  Apparently $X$ must be infinite, and so we may enumerate $X=\{x_i:i<\omega\}$  where the $x_i$'s are distinct. 

Let $C_0=X$.  

Suppose $n\geq 1$ and non-degenerate continua $C_0\supseteq C_1\supseteq ... \supseteq C_{n-1}$ have been defined.

Let  $x^*$ be the element of $C_{n-1}$ with least subscript. 

Let $x^{**}$ be the element of $C_{n-1}$ with least subscript greater than $x^*$'s. 

Let $\mathcal U_n=\{U\subseteq X:U \text{ is open, }x^*\in U\text{, and }x^{**}\notin \overline U\}.$ Since $X$ is Hausdorff, $\mathcal U_n\neq\varnothing$. Let $$\mathcal C_n=\{C\subseteq C_{n-1}:x^{**}\in C,\;C\text{ is connected, and }(\exists U\in \mathcal U_n)(C\cap U=\varnothing)\}.$$Let $C_n=\overline{\bigcup \mathcal C_n}$. Then $C_n$ is a continuum, and is non-degenerate because some elements of $\mathcal C_n$ are non-degenerate.  This is true because compactness and normality of $X$ implies the quasi-component of $x^{**}$ in $C_{n-1}\setminus U$, $U\in \mathcal U_n$, is connected, and this quasi-component must meet $\partial U$ in order for $X$ to be connected. (Choice is not needed to prove normality, nor is it needed to prove these quasi-components are connected.)

Continuing in this manner, we construct a nested sequence $(C_n)$ of non-empty compact sets.  Their intersection must be non-empty.  But on the other hand we ensured each point of $X$ is eventually not in $C_n$. Contradiction. $\blacksquare$


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Related: In 2013 Horst Herrlich and  Kyriakos Keremedi proved that "[Connected separable metric spaces need not have continuum size in ZF][1]". 

My follow-up question: *Is every separable metric continuum equinumerable with the reals, in ZF?*  


  [1]: https://www.sciencedirect.com/science/article/pii/S0166864113004409