[Math] Example of a submanifold $S\subseteq M$ that is an immersed submanifold is more than one way

Question

Known uniqueness results say that an embedded submanifold has a unique smooth structure making it an embedded submanifold with the subspace topology, and immersed submanifolds have a unique smooth structure making them immersed if we have a prior fixed topology.

This seems to imply that there is an instance of some immersed submanifold $S\subseteq M$ with a given topology and smooth structure, where it's possible to make $S$ an immersed submanifold with different smooth structure if we're also allowed to endow it with a different topology.

Is there a known example of such a submanifold? My definition is the stricter one that a smooth manifold is the image of an injective immersion. If no example exists, I wouldn't mind relaxing the definition to the image of just an immersion.

Answer 1

The key is to find a set that is the image of two different injective immersions that induce different topologies on it. Here's a hint:

$\qquad$

EDIT: For an example where the two submanifolds are not homeomorphic, try this:

In one case, the submanifold has two connected components, neither of which is compact. In the other, there are three components, one of which is compact.