Cover a table with a tablecloth, crumple it up in the middle (while still leaving the edges hanging over the edge of the table), then stab the folds with a pin. You will almost surely poke an odd number of holes in the tablecloth, because one end of the pin is above the tablecloth, the other is below, and each hole indicates one instance of the pin changing sides (the only exception is if your stab lies perfectly tangent to a fold of the tablecloth).

I suspect that this generalizes. I hope that something like this is true:

Let $X$ be a subset of $\mathbb{R}^n \times \mathbb{R}^m$ that is homeomorphic to $\mathbb{R}^n$. For any $r \in \mathbb{R}^n$, let $D(r) = \{ s \in \mathbb{R}^m | (r, s) \in X \}$. Suppose $D(r)$ is never empty for any $r \in \mathbb{R}^n$. Then $D(r)$ almost always (with respect to Lebesgue measure) has an odd number of elements.

In other words, we have crumpled an $n$-dimensional tablecloth into $m$ additional dimensions (while still leaving it hanging over the edges of our $n$-dimensional table), then stabbed it with an $m$-dimensional pin, hopefully in an odd number of places.

Is this a known result? It smells a lot like Sperner's Lemma (which contains a similar statement about oddness), but I'm not entirely sure how.