Suslin's problem is:



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> Given a complete dense linear order without endpoints, if it has the ccc must it be isomorphic to $\mathbb{R}$?
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The answer is that it's independent of ZFC. The related question:



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> Given a complete dense linear order without endpoints, if it's separable must it be isomorphic to $\mathbb{R}$?
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has a positive answer under ZFC. Now consider the following analogous questions:



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> Given a non-trivial separative forcing poset, if it has the ccc must it have size at most continuum?
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The answer to this is no, for example the Cohen forcing that adds more than continuum-many reals is ccc but has size greater than continuum. So what about:



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> **Given a non-trivial separative forcing poset, if it's separable (i.e. has a countable dense set) must it have size at most continuum?**
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A problem about posets similar to Suslin's problem
Amit:


If ${\mathbb P}$ is a non-trivial separative partial order, and it is countable, an easy argument (back-and-forth) shows that it is forcing isomorphic to Cohen forcing. (This is an exercise in Chapter VII of Kunen's book, I believe, and it can be found in a few other sources, such as the appropriate chapter of the "Handbook of Boolean Algebras".)


It follows that if ${\mathbb P}$ is non-trivial, separative, and admits a countable dense subset, then ${\mathbb P}$ is again Cohen forcing. Now, ${\mathbb P}$ embeds into its Boolean completion which must then coincide with the Boolean completion of Cohen (Borel sets/meager) and therefore has size ${\mathfrak c}=|{\mathbb R}|$. This gives ${\mathfrak c}$ as an upper bound for $|{\mathbb P}|$.


Here is a cute application: Fix $\epsilon>0$, and let ${\mathbb P}^\epsilon$ be the collection of open subsets $p$ of ${\mathbb R}$ that are a finite union of intervals with rational end-points and such that the sum of their lengths (the Lebesgue measure of $p$) is less than $\epsilon$. The generic object gives us an open set that covers the ground model reals, and has measure $\epsilon$. Since ${\mathbb P}^\epsilon$ is countable, it is Cohen forcing. Since $\epsilon$ is arbitrary, it follows that adding a Cohen real makes the set of ground model reals have measure zero.




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**Edit:** As Andreas Blass pointed out, one can actually prove the upper bound $|{\mathbb P}|\le{\mathfrak c}$ directly. This actually gives us a way of *proving* the characterization of Cohen forcing.


