Began by showing John Nash's PhD thesis in which he uses Brouwer's theorem to prove the existence of (what are now called) mixed Nash equilibria.
Defined what is means for two spaces to be homotopy equivalent. Showed that any two homeomorphic spaces are homotopy equivalent. Also showed that the space C\{0} of non-zero complex numbers is homotopy equivalent to a circle.
Stated a major theorem.
Theorem. Homotopy equivalent spaces have the same Euler characteristic.
Illustrated this theorem with a few examples.
Stated Brouwer's fixed-point theorem: any continuous function f:Δn ---> Δn has a fixed point.