Study of geometric properties preserved under continuous deformations.
Topological invariants are properties of spaces that don’t change under homeomorphisms. They help distinguish spaces that look similar but are topologically different.
They allow mathematicians to classify and compare spaces, solve puzzles, and prove theorems.
The Euler characteristic, which relates vertices, edges, and faces of a shape:
\[
\chi = V - E + F
\]
The fundamental group, which captures how loops in a space can be deformed.
A donut and a coffee cup both have one hole, so their genus is one.
A sphere has genus zero, distinguishing it from a donut.
Topological invariants are features of spaces that survive stretching and bending.