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In algebraic geometry, flat morphisms are usually associated with the intuition that they have „continuously varying fibers”. Is there a notion in topology formalizing the same intuition? Consider for example a map $p \colon P \to X$ of good (locally compact Hausdorff, say) topological spaces and assume for the moment that it is proper (i.e. has compact fibers). If the fibers vary continuously, then the following property should be true for example:

(*) If $f \colon P \to \mathbf{R}$ is a continuous function, then the induced function $$\widetilde{f} \colon X \to \mathbf{R}, \qquad \widetilde{f}(x) := \sup_{q \in p^{-1}(\{x\})}f(q)$$ is also continuous.

This is true for example if the fibers are constant, i.e. if $p$ is a trivial bundle $p \colon X \times F \to X$. In that case, the map $f \colon X \times F \to \mathbf{R}$ induces a continuous map $X \to C(F, \mathbf{R})$, the space of continuous functions equipped with the compact-open topology (which is the topology of uniform convergence) and the map $\widetilde{f}$ factors as $X \to C(F, \mathbf{R}) \xrightarrow{\sup} \mathbf{R}$.

But there are also examples where the fibers are not (locally) constant: For example the map $\mathbf{C} \to \mathbf{C}$, $z \mapsto z^2$ has continuously varying fibers in the intuitive sense (as $z$ approaches the origin, the two points in the fiber get closer to each other). This is also a flat map in algebraic geometry.

For an example which is not flat in the intuitive sense and where also the above property is clearly wrong, consider the first projection $$p \colon \{(x,y) \in \mathbf{R} \times [0,1] \mid x \cdot y = 0 \} \to \mathbf{R}.$$ (This example is of course also inspered by algebraic geometry.)

I should perhaps add that I am aware of the related questions The topological analog of flatness? and Flatness in Algebraic Geometry vs. Fibration in Topology. But I am interested in the purely point-set-topological setting and in fact I care specifically about the property (*) (I have a specific example of a continuous map $p$ in mind for which I hope (*) to be true and I'm trying to gain some intuition.)

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  • $\begingroup$ I think the only reasonable topological analog is that of an open map. Note that open maps are closed under base-change in the topological category. It's a fun question if open proper maps satisfy your condition * and I wish I had energy to think about it right now... $\endgroup$
    – Dustin Clausen
    Commented Sep 8, 2021 at 18:23
  • $\begingroup$ @DustinClausen Thank you so much! I could verify my property (*) for proper surjective open maps of topological spaces (as long as the definition of „proper” includes „closed”, I don't need any condition on the spaces). I also found a not-so-easy theorem which guarantees that my map $p$ is open, so my initial problem is solved. $\endgroup$
    – Jakob Werner
    Commented Sep 11, 2021 at 15:59
  • $\begingroup$ @DustinClausen I would still be interested in why you think that openness is the only reasonable topological analog of flatness. $\endgroup$
    – Jakob Werner
    Commented Sep 11, 2021 at 16:01
  • $\begingroup$ @JakobWerner Great, I'm glad my comment was helpful and you could solve your problem. About why I think openness is the only reasonable topological analog of flatness, I'm not really sure how to make precise this feeling of mine. But in several examples I've seen the notion of openness play the same role as the notion of flatness. And the only general topological consequence of flatness that I'm aware of is openness. $\endgroup$
    – Dustin Clausen
    Commented Sep 12, 2021 at 11:23

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