What does "an immersed sub manifold is in general not a submanifold as a subset" mean?

Question

According to Wikipedia https://en.wikipedia.org/wiki/Submanifold : An immersed submanifold of a manifold M is the image S of an immersion map f: N → M; in general this image will not be a submanifold as a subset, and an immersion map need not even be injective (one-to-one) – it can have self-intersections.

I am not understanding what this last sentence means. Could someone please elaborate on what "submanifold as a subset" means?

Answer 1

The thing is that in general, even a 1-1 immersion (one for which no self-intersections take part) can present even more horrendous problems. What the "... submanifold as a subset ..." part means is that the image of a 1-1 immersion may have a subspace topology different than the one induced by the immersion, i.e the 1-1 immersion may not be a homeomorphism onto its image, which is one of the equivalent characterizations of a regular or embedded submanifold.
So, the problem is a little more subtle than the thing with the self-intersections.
As an example, if you embedd $\mathbb{R}$ into $\mathbb{R}^2$ injectively as in Figure 1-9 pp. 16 of $\textit{Differential Topology}$, by Guillemin & Pollack, you certainly don't have self-intersections, but your immersion fails to be homeomorphism because it is not a proper map. Just take a compact neighborhood around the special point and its preimage is not compact.
Another well-known example is the line with irrational slope in the torus, as a 1-1 immersion $\mathbb{R}\rightarrow \mathbb{T}^2$ (Example 5.3 pp. 95 in $\textit{Introduction to Smooth Manifolds}$ by J. M. Lee), where the image of the 1-1 immersion is even dense in the ambient!

Answer 2

For example the sign $\infty$ (or $8$ if you prefer) is the image of an immersion $S^1 \to\Bbb R^2$, but it is not a submanifold, because the cross point has no neighborhood homeomorphic to $\Bbb R^n$ in the subspace topology (though all other points have, with $n=1$).