How to intuitively visualize open maps

Question

I intuitively think of continuous maps as functions whose behavior as they approach a point is the same as their behavior at the point. How should I intuitively think of open maps?

Answer 1

If you're ok with infinitesimals, either informally or as formalized in, say, Robinson's non-standard analysis, that can provide some geometric intuition.

Thus:

"$f$ is continuous at $x_0$ when, for any $x$ infinitesimally close to $x_0$, $f(x)$ is infinitesimally close to $f(x_0)$."

(The formal version of this appears in Robinson, Non-standard Analysis, p.66. The technical statement includes a fine point or two.)

So if you picture $x_0$ as a dot, and all the $x$'s infinitesimally close to $x_0$ as a "fuzzy dot", this says that the image of the fuzzy dot is contained in a fuzzy dot around $f(x_0)$. In other words, $f$ can't "blow up the infinitesimal neighborhood of $x_0$ to finite size".

Being open is the flip side: images of fuzzy dots "remain fuzzy". For example, consider $f(x)=x^2$ on $\mathbb{R}$. This is open except at $x=0$. The image of the fuzzy dot centered at 0 is "folded over" by $f$, so it looks fuzzy on the right but has a sharp edge on the left.