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dfeuer
dfeuer
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Let $X$ be a compact, connected Hausdorff space with at least two points.

In $\mathrm{ZF}+\mathrm{AC}_\omega(\mathbb R)$, $X$any countable compact Hausdorff space is metrizable, and from this it can be shown that $X$ is uncountable.

In $\mathrm{ZF}$, however, the space maythat result does not be metrizablehold. Does anyone know if it's still possible to prove that it is uncountable?

Let $X$ be a compact, connected Hausdorff space with at least two points.

In $\mathrm{ZF}+\mathrm{AC}_\omega(\mathbb R)$, $X$ is metrizable, and from this it can be shown that $X$ is uncountable.

In $\mathrm{ZF}$, however, the space may not be metrizable. Does anyone know if it's still possible to prove that it is uncountable?

Let $X$ be a compact, connected Hausdorff space with at least two points.

In $\mathrm{ZF}+\mathrm{AC}_\omega(\mathbb R)$, any countable compact Hausdorff space is metrizable, and from this it can be shown that $X$ is uncountable.

In $\mathrm{ZF}$, however, that result does not hold. Does anyone know if it's still possible to prove that it is uncountable?

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dfeuer
dfeuer
  • 511
  • 3
  • 12

Is it possible to prove in ZF that a non-trivial compact connected Hausdorff space is uncountable?

Let $X$ be a compact, connected Hausdorff space with at least two points.

In $\mathrm{ZF}+\mathrm{AC}_\omega(\mathbb R)$, $X$ is metrizable, and from this it can be shown that $X$ is uncountable.

In $\mathrm{ZF}$, however, the space may not be metrizable. Does anyone know if it's still possible to prove that it is uncountable?