Nonarchimedean analysis studies the properties of convergence in spaces that do not satisfy the Archimedean property. Examples of such spaces include the $p$-adic numbers and hyperreal and surreal numbers.
In $\mathbb{R}$ with the usual absolute value $|\cdot|$, the Archimedean property is the statement that if $0<x<y$, there is an $n\in\mathbb{N}^+$ such that $nx >y$; in a general ordered field, this is replaced with the condition $\underbrace{x+\cdots +x}_{n}>y$. A non-Archimedean space is a space that does not satisfy this property. The most familiar example is the $p$-adic numbers under the valuation metric.
