What is the difference between "immersion" and "embedding"?

Solution 1:

Understandably there are a lot of answers, but if you still have any further questions maybe this will help.

An embedding of a topological space $X$ into a topological space $Y$ is a continuous map $e \colon X \to Y$ such that $e|_X$ is a homeomorphism onto its image.

Both the Klein bottle ($f \colon I^2 / \sim \to \mathbb{R}^3$) is not embedded into $\mathbb{R}^3$, because it has self-intersections; this means that the immersion of the Klein bottle is not a bijection, hence not a homeomorphism, so not an embedding.

As I understand it, an immersion simply means that the tangent spaces are mapped injectively; i.e. that the map $D_p f \colon T_pI^2 \to T_{f(p)}\mathbb{R}^3$ is injective. In the Klein bottle example, at the self-intersection, any point of intersection has two distinct tangent planes, hence this map is injective.

I hope this makes some sense!

Solution 2:

Basically an abstract surface has, at every point two independent directions along the surface. Or even better, there is an entire circle's worth of rays coming out of each point.

An immersion is, roughly, a map of the surface into a bigger manifold (such as $\mathbb R^n$) where there are still two dimensions worth of rays emanating out of each point. So for the usual immersion of a Klein bottle into $\mathbb R^3$, at the circle of self-intersection, each sheet still retains its two dimensional character. So it is an immersion. If you were to instead map the Klein bottle into $\mathbb R^3$ by mapping everything to a point, that would not be an immersion.

Solution 3:

via Wikipedia on Immersions

An immersion is precisely a local embedding – i.e. for any point x ∈ M there is a neighbourhood [sic], U ⊂ M, of x such that f : U → N is an embedding, and conversely a local embedding is an immersion.

So, an immersion is an embedding, i.e. an isomorphic (homeomorphic) copy, at each point, and vice versa, though the entire image may not be a homeomorphic copy.

But, later, the same article says:

If M is compact, an injective immersion is an embedding, but if M is not compact then injective immersions need not be embeddings; compare to continuous bijections versus homeomorphisms.

I wish I could give an example of a non-compact imbedding/embedding, or continuous bijections versus homeomorphisms, but though I understand the two ideas, roughly, I am not sure what conditions make them conflict...