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Real line is separable.

Then, why is $\omega_1$ topological space not separable?

IF this is true, doesn't this settle continuum hypothesis?

Also, does the field of real numbers have anything to do with topological spaces? (Just for curiosity.)

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The order topology on $\omega_1$ has no direct relationship to the Euclidean topology on $\Bbb R$, so there’s no reason for the separability of one to suggest the separability of the other, let alone to imply it. The order topology on $\omega_1$ is not separable for the simple reason that if $A$ is a countable subset of $\omega_1$, then there is a $\beta\in\omega_1$ such that $\alpha<\beta$ for every $\alpha\in A$. This means that if $\gamma\in\omega_1$ and $\gamma>\beta$, the interval $(\beta,\omega_1)$ is an open neighborhood of $\gamma$ that is disjoint from $A$, and therefore $\gamma$ is not in the closure of $A$. Thus, no countable subset of $\omega_1$ can be dense in $\omega_1$.

None of this involves the continuum hypothesis in any way.

‘Anything to do with’ is a pretty vague description. If you’re asking whether the properties of $\Bbb R$ as an algebraic field play a significant rôle in topology, I’d say not, but that may be partly a reflection of my particular topological interests. Be that as it may, it’s fair to say that the order properties of $\Bbb R$ are much more important topologically.

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In some models where the axiom of choice fails we have that $\omega_1$ is a countable union of countable ordinals. This means that indeed $\omega_1$ is separable in such model.

Thus, assuming $\omega_1$ is separable is inconsistent with ZFC, but is fairly consistent with ZF. In fact it settles the question whether or not it is a singular cardinal!

Now we can consider the continuum hypothesis in two formulations in ZFC:

  • The Continuum Hypothesis (CH): Every uncountable set of real numbers has size continuum.
  • The Aleph Hypothesis (AH): The continuum is of size $\aleph_1$.

While in ZFC these are equivalent they are no longer equivalent in ZF. The assumption that $\omega_1$ is singular automatically implies that AH is false. It need not imply that CH is false, but I'm not sure that has an answer so far.

I should also point out that given an uncountable set one can consider the discrete topology. It is a locally compact metric space, and by that it is a bit closer to the real line than $\omega_1$ (where both are simply linearly ordered). Uncountable discrete spaces are never separable since every dense set would include all points.

For Further Reading:

  1. The Continuum Hypothesis and Countable Unions (MathOverflow),
  2. Relationship between Continuum Hypothesis and Special Aleph Hypothesis under ZF
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