At the end of the Wikipedia article on Deformation Retract, there is the following sentence:

Two spaces are homotopy equivalent if and only if they are both deformation retracts of a single larger space.

I was wondering if this has some meaning in Algebraic Geometry. For instance, consider the situation of a flat surjective map of complex schemes $f:X\to S$, where $S$ is a smooth curve, e.g. the formal disk.

**Question 1**. Are the fibers of $f$ homotopy equivalent? Is every fiber $X_s$ a deformation retract of the total space $X$?

I tried to show, first, that any fiber is a retract of the total space, but I am unsure about my solution, and in any case it is weaker than deformation retract. But it goes as follows: if $X_0$ is a particular fiber of $f$, we can define a retraction $r:X\to X_0$ as the identity on $X_0$, and by $$r(x)=\{x\}^-\cap X_0,\,\textrm{for }x\in X\setminus X_0.\,\textrm{(this is the flat limit I guess)}$$

**Question 2**. In the topological category, say we have a topological fiber bundle $X\to S$. Are the fibers homotopy equivalent?

Thanks for any clue on this!