Euler's Gem: The Polyhedron Formula and the Birth of Topology

This book is a very well written and easy to read introduction to why topology exists. The central theme of the book is Euler's polyhedron formula, which is so simple that anyone who knows how to count should feel comfortable with it. Richeson follows the emergence of the deep idea that lurks behind this formula as the mathematics community came to grips with it. Proofs were refined, the formula was generalized, and eventually, an entirely new discipline emerged.

There are occasional dives into "real" maths, but nothing even remotely complicated. Well worth reading, even if you are already aquainted with topology.

"V - E + F = 2" here serves as a hook for "a history and celebration of topology." Authoritative, and inclusive of quite a few proofs.