Regular homotopy allows self-intersections but never rank drop

A regular homotopy $H:M\times [0,1]\to N$ of immersions requires every slice $H_t$ to be immersive (Regular homotopy of immersions, Smooth families of maps and their evaluation maps): the rank of $d{H}_t$ is $\dim M$ at every point of every slice (Immersions, submersions, and constant-rank maps), and no slice may contain a point of rank drop, a cusp of the parametrised family or a point whose derivative degenerates.

Self-intersections, by contrast, are permitted: an immersion may identify two distinct points $x\mathrm{\ne }y$ with $H_t(x)=H_t(y)$ as long as the differential is injective at each point. Their tangent images may coincide; transversality is not required by the immersion condition. During an eversion of $S^2$ in ${\mathbb{R}}^3$ the slices must acquire self-intersections and cannot be embeddings (Sphere eversion cannot be an isotopy through embeddings), while no slice may have a rank drop, so the two phenomena are logically independent.

For a fixed smooth map, the set of source points where its differential has full rank is open (The immersion and submersion loci are open). This is a statement about the source, rather than about a topology on a space of maps. A smooth family is a regular homotopy exactly when every slice is an immersion; immersive slices for $t<1$ do not guarantee an immersive final slice.