Can manifold subsets always be made into submanifolds?

Question

Related question: Are manifold subsets submanifolds?

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Assume all manifolds, topological or smooth discussed here have dimensions and do not have boundary.

Let $A'$ and $B'$ be sets with $A' \subseteq B'$.

Question A: Are these correct?

1. As far as I know, all sets can be made into topological spaces.

2. By (1), make the sets $A'$ and $B'$ into, respectively, the topological spaces $A$ and $B$.

3. I haven't thought about whether there are some sets that cannot be given topological spaces that enable them to become smooth or topological manifolds, but as far as I know, some topological spaces cannot be made into smooth manifolds or even topological manifolds...such as ones that aren't Hausdorff I guess.

4. By (3), assume $A$ and $B$ from (2) can be made into smooth manifolds $(A,\mathscr A)$ and $(B,\mathscr B)$ where $\mathscr A$ and $\mathscr B$ are smooth atlases.

5. By (4) and the above related question, $(A,\mathscr A)$ is not necessarily a (regular/an embedded) smooth submanifold of $(B,\mathscr B)$ or even an immersed smooth submanifold.

Question B: Does there exist a smooth atlas $\mathcal A$ where $(A,\mathcal A)$ becomes a smooth submanifold of $(B,\mathscr B)$?

• I hope Question $B$ is equivalent to both

• (C) "If a topological subspace can become a smooth manifold, then can it become a smooth submanifold?"

• (D) "Can smooth manifold subsets $(N,\mathscr N)$ of smooth manifolds $(M, \mathscr M)$ always be made into smooth submanifolds $(N,\mathcal N)$ of $(M, \mathscr M)$?"

• If one of (1)-(5) is wrong, then $B$, $C$ or $D$ could be meaningless or, if meaningful, not equivalent to the other meaningful ones. Please answer the meaningful ones among (B),(C) and (D), and please point out which are equivalent or not.

• Not concerned about uniqueness at this point. You can say something about uniqueness if you want.

Answer 1

Your statements in Question A are all essentially correct. Whether or not you can put a manifold topology on a set depends only on the cardinality of the set. For sets with cardinality greater than that of $\mathbb R,$ the answer depends on your axioms:

• If your definition of manifold includes second-countability, then any set strictly larger than $\mathbb R$ cannot be made a manifold.
• If not, every set can be made a 0-dimensional manifold. (If you insist on positive dimension, I think by assuming the axiom of choice you can prove that all sets with cardinality at least that of $\mathbb R$ can be made manifolds.)

Regardless, I think this point is a tangent from your main concern:

Question B. You can answer this without needing to think about smooth structure at all: If $A$ has an atlas making it a smooth submanifold of $B,$ then $A$ is a topological submanifold of $B.$ This makes things much easier, since there is no arbitrary choice to be made when talking about topological submanifolds: a subset $A \subset B$ is a topological submanifold if and only the induced topology makes $A$ a topological manifold.

So, if $A$ has some atlas making it a smooth submanifold, then $A$ is a topological manifold in the induced topology. Thus any subset $A \subset B$ which is not a topological manifold in the induced topology (e.g. one that is not locally Euclidean) provides a counterexample.