Squish, Crease, Tear: An Intuition for Open and Continuous Maps in Topology

When we first encounter the definition of a continuous function in general topology, it often feels completely backward: Why is a map defined as continuous if the pre-image of every open set is open?

Shouldn't a structure-preserving map send open sets to open sets (open map) instead?

Most texts warn us not to confuse the two, throw a couple of counterexamples, and move on.

Here is an essay building an intuition for why this definition works the way it does and why "pushing forward" fails while "pulling back" works to ensure continuity.