Intuition of Immersed versus Embedded Submanifolds

Question

The definitions I read in Lee's Smooth Manifolds is:

Embedded Submanifold: $S\subset M$ is an embedded submanifold if $S \to M$ is an embedding.

Immersed Submanifold: $S\subset M$ is an immersed submanifold if $S \to M$ is an injective immersion.

Thus, an immersed submanifold with the subspace topology is an embedded submanifold. It is also true for an immersed submanifold $S$ that all $U\subset S$ that are open in the subspace topology are open in $S$ (in the immersed submanifold topology).

How does one think of immersed versus injective sub manifolds? Intuitively how do they differ?

Answer 1

Injective immersions that are not embeddings can be seen as an exotic type of injective immersion which is only possible when the spaces or map involved are not sufficiently nice. After all, an injective immersion $i:M\to N$ must be an embedding if

• $M$ is compact
• $M$ and $N$ have the same dimension (and $\partial M=\varnothing$)
• $i$ is open or closed
• $i$ is proper

(This is Proposition 4.22 in Lee.) You can make injective immersions that are not embeddings by taking manifolds that are not closed and folding their open edges to "touch" the manifold. For example, take an open interval and fold it over itself to make a lemniscate, as shown below. The image of this map is not a manifold.