In SGA 4.5 (Arcata V.1) Deligne writes: 

> Let $X$ be a complex analytic variety and $f:X\rightarrow D$ map $X$ into the
> disk.  Write $[0,t]$ for closed line segment with extremities 0 and $t$ in
> $D$, and $]0,t]$ for the half open segment.  If $f$ is smooth (lisse) the
> inclusion $j:f^{-1}(]0,t])\rightarrow f^{-1}([0,t])$ is a homotopy 
> equivalence; one can push the special fiber $X_0=f^{-1}(0)$ into
> $f^{-1}([0,t])$.

I think that last $[0,t]$ at the end of the sentence is a typo and should say $]0,t]$.  Is that right?

Of course this motivating example from analytic geometry is not itself exactly scheme theory but the comparison is clear.  Anyway Deligne concludes this passage with: "one can express this construction imagistically by saying, for a smooth  map, the general fiber swallows (avale) the special fiber."  He uses the same image to motivate proper base change by comparison with point set topology.

I think the idea of the image, in scheme terms, is that the fiber at any point always maps to the fiber at any specialization -- but in general the special fiber does not map back to the generalization.  And when it does map back he says the general fiber swallows the special fiber.   Am I right to think of it that way?