History of Mathematics

Definition 1.1Let f:X⟶Yand f′:X⟶Ybe two continuous maps of the space X into the space Y. We say that f is homotopic to f′, written f≃f′, if there exists a continuous map F:X×I⟶Ywhere I denotes the interval , such that F(x,0)=f(x)and F(x,1)=f′(x)The map F is called a homotopy between f and f′. If f≃f′ and f′ is a constant map, we call f null-homotopic. In other words,...

Incompleteness Mathematicians love simplicity. Back around 1900, their dream was to create a world of math that was cleanly split in two. True statements would all be provable from axioms. The false statements, none of them...