What is the most beautiful definition you've encountered?

[olliemath]

Continuity in the way I first heard it:

A funtion $$f:A\subseteq\mathbb{R}\rightarrow\mathbb{R}$$ is continuous if, for each $$x\in A$$ and each $$\varepsilon>0$$, we can find some $$\delta>0$$ such that $$|f(x)-f(y)|<\epsilon$$ whenever $$|x-y|<\delta$$.

I know a lot of people tend not to like it when they first see it, but it was the first time I saw a mathematician take something so intuative then transform it into a solid mathematical form. That gave me a love of analysis/topology that I still hold. An alternative for me would possibly be the definition of the fundamental group.

 

[mathwonk]

what about the definition where f^-1(U) is open whenever U is open.

 

[olliemath]

I like the classical form (I think because I have the memory "wow, maths can be beautiful" associated to it, rather than its intrinsic genius) but I can see why you like the more topological version.

 

[mutton]

compactness: every open cover has a finite subcover

 

[EternalVortex]

I like the definition of NP the best. Easy to verify, hard to solve.

 

[CRGreathouse]

I like the definition of NP the best. Easy to verify, hard to solve.

That's a good one.

 

[tim_lou]

All those cohomology business. It's hard to believe how much antisymmetry (of simplexes, tensor products..etc) gives you... Stokes theorem, De Rham's theorem and what not...even though I haven't fully understood them yet.

 

[tim_lou]

The definition of Lebesgue integral as well. To understand how Riez representation theorem falls right out of it is amazing.