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Let $M$ be an open topological manifold. Must the number of non-diffeomorphic smooth structures on $M$ be either countable or continuum? Could it be something in between when the continuum hypothesis fails?

Edit:

1. By "open manifold" I mean "not necessarily closed", although it seems the standard meaning is "not closed".

2. As pointed out in the comments, we should focus on second-countable manifolds to make the question more interesting.

3. There are examples showing the number can be finite, countably infinite, or continuum.

   • Finite: any closed manifold of dimension other than $4$.

   • Countably infinite: for example $S^2\times S^2$. Ryan Budney mentions in a comment here the conjecture that every smoothable closed 4-manifold has countably infinitely many smooth structures.

   • Continuum: $\mathbb{R}^4$

Let $M$ be a manifold. Must the number of non-diffeomorphic smooth structures on $M$ be either countable or continuum? Could it be something in between when the continuum hypothesis fails?

Edit:

1. By manifold I mean a connected Hausdorff second-countable locally Euclidean space (so without boundary). I'm not sure how dropping Hausdorffness or allowing boundary would change the problem, but let's keep things simple.

2. Although I'm asking about second countable manifolds, it seems the problem remains nontrivial without this assumption. As pointed out in the comments, the long line has $2^{\aleph_1}$ non-diffeomorphic smooth structures, but $2^{\aleph_1}$ is not stricly between countable and continuum...

3. WLOG we may focus on open (non-compact) manifolds, since the number of smooth structures on a closed manifold is always countable, in fact always finite except in dimension $4$.

4. There are examples showing the number can be finite, countably infinite, or continuum.

   • Finite: any closed manifold of dimension other than $4$.

   • Countably infinite: for example $S^2\times S^2$ (I couldn't find an open example). Ryan Budney mentions in a comment here the conjecture that every smoothable closed 4-manifold has countably infinitely many smooth structures.

   • Continuum: $\mathbb{R}^4$

Let $M$ be an open topological manifold. Must the number of non-diffeomorphic smooth structures on $M$ be either countable or continuum? Could it be something in between when the continuum hypothesis fails?

Edit:

1. By "open manifold" I mean "not necessarily closed", although it seems the standard meaning is "not closed".

2. As pointed out in the comments, we should focus on second-countable manifolds to make the question more interesting.

3. There are examples showing the number can be finite, countably infinite, or continuum.

   • Finite: any closed manifold of dimension other than $4$.

   • Countably infinite: for example $S^2\times S^2$. Ryan Budney mentions in a comment here the conjecture that every smoothable closed 4-manifold has countably infinitely many smooth structures.

   • Continuum: $\mathbb{R}^4$

Let $M$ be an open topological manifold. Must the number of non-diffeomorphic smooth structures on $M$ be either countable or continuum? Could it be something in between when the continuum hypothesis fails?

Edit:

1. By "open manifold" I mean "not necessarily closed", although it seems the standard meaning is "not closed".

2. As pointed out in the comments, we should focus on second-countable manifolds to make the question more interesting.

3. There are examples showing the number can be finite, countably infinite, or continuum.

   • Finite: any closed manifold of dimension other than $4$.

   • Countably infinite: for example $S^2\times S^2$. Ryan Budney mentions in a comment here the conjecture that every smoothable closed 4-manifold has countably infinitely many smooth structures.

   • Continuum: $\mathbb{R}^4$

Let $M$ be an open topological manifold (Edit: as pointed out in the comments, we should focus on second-countable manifolds to make the question more interesting). Must the number of non-diffeomorphic smooth structures on $M$ be either countable or continuum? Could it be something in between when the continuum hypothesis fails?

Remark: there are examples showing the number can be finite, countably infinite, or continuum.

Finite: any closed manifold of dimension other than $4$.

Countably infinite: for example $S^2\times S^2$. Ryan Budney mentions in a comment here the conjecture that every smoothable closed 4-manifold has countably infinitely many smooth structures.

Continuum: $\mathbb{R}^4$

Let $M$ be an open topological manifold. Must the number of non-diffeomorphic smooth structures on $M$ be either countable or continuum? Could it be something in between when the continuum hypothesis fails?

Edit:

1. By "open manifold" I mean "not necessarily closed", although it seems the standard meaning is "not closed".

2. As pointed out in the comments, we should focus on second-countable manifolds to make the question more interesting.

3. There are examples showing the number can be finite, countably infinite, or continuum.

   • Finite: any closed manifold of dimension other than $4$.

   • Countably infinite: for example $S^2\times S^2$. Ryan Budney mentions in a comment here the conjecture that every smoothable closed 4-manifold has countably infinitely many smooth structures.

   • Continuum: $\mathbb{R}^4$