I am reading Guillemin and Pollack.
The definition of an immersion they give is:
$f: X \to Y$ is an immersion at $x \in X$ if $df_x : T_x(X) \to T_y(Y)$ is injective. If $f$ is an immersion $\forall x \in X$, then we say that $f$ is an immersion.
So apparently the map from the circle to the figure 8 is an immersion, as they state on the next page. But what about the critical point in the mapping $f$? There is no tangent space defined here, correct? So then how could $f$ possibly be an immersion?
Also, is there a simple example of something that isn't an immersion that will help me remember this definition?