Update. I have asked this on MO, but have not yet received an answer.

Proposition. The quotient map associated to a topological foliation (projecting to the leaf space) is open.

However the fibers of an open map from a topological manifold need not foliate its domain.

Example. Consider the multiplication map $\mathbb R^2\to \mathbb R,(x,y)\mapsto xy$. It is open, but I think its fibers do not give a topological foliation of the plane because the fiber over zero is the union of the axes, which does not look like parallel lines locally about the origin.

In the above example, the problem is at an isolated point. That leads me to wonder how far the fibers of an open map can be from "generically" foliations.

Definition. Let $X$ be a topological manifold. A partition of $X$ is generically a foliation of $X$ if it restricts to a foliation of an open dense subspace (which is also a topological manifold).

Example. I think the fibers of the multiplication maps give a generic foliation of the plane as they foliate the punctured plane.

I'm fairly certain a map (set function even) $f:X\to Y$ of topological spaces is open iff for every convergent net $y_\alpha\to y$ we have $f^{-1}(y)\subset\lim \left\{ f^{-1}(y_\alpha ) \right\}$, so the fibers of an open map are "stuck together" in this sense. I would like to understand exactly how different the latter sense is from the fibers actually being generically foliating.

Question. What are some examples of a (continuous) open map from a topological manifold $X$ whose fibers are not generically a foliation of $X$? What are some examples if we moreover assume the map is a quotient map?

Remark. I am looking for maximally "geometric" (i.e minimally "fractal") examples.

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    $\begingroup$ Any references for topological foliations? $\endgroup$ – PtF Jan 6 '18 at 16:04
  • $\begingroup$ @PtF not really a reference but see Milnor's Foliations and Foliated Vector Bundles for a definition. Essentially the foliated atlas is required to have homeomorphic charts without smoothness assumptions. $\endgroup$ – Arrow Jan 6 '18 at 16:13

Some examples of open maps which are not generically foliations are given by dimension-raising continuous open maps from manifolds to cells (it's hard to imagine that such maps exist...), see my answer here and more examples and references for instance here:

J.J.Walsh, Monotone and open mappings on manifolds, I, Transactions of the American Mathematical Society Vol. 209 (Aug., 1975), pp. 419-432.

  • $\begingroup$ Thanks for the answer! Those seem like unsightly creatures. I will wait for some more answers in hopes of something I can visualize. $\endgroup$ – Arrow Jan 28 '18 at 17:26
  • $\begingroup$ If they are unsightly or or not is a matter of opinion. Compare: imdb.com/title/tt0734568, "it comes recommended". $\endgroup$ – Moishe Kohan Jan 29 '18 at 4:11

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