In general topology, a pretopological space is a generalization of the concept of topological space. A pretopological space can be defined as in terms of either filters or a preclosure operator. The similar, but more abstract, notion of a Grothendieck pretopology is used to form a Grothendieck topology, and is covered in the article on that topic. Let X be a set. A neighborhood system for a pretopology on X is a collection of filters N(x), one for each element x of X such that every set in N(x) contains x as a member. Each element of N(x) is called a neighborhood of x. A pretopological space is...

In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site. Grothendieck topologies axiomatize the notion of an open cover. Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology. This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme. It has...

In mathematics, specifically order theory, a well-quasi-ordering or wqo is a quasi-ordering such that any infinite sequence of elements x 0 {\displaystyle x_{0}} , x 1 {\displaystyle x_{1}} , x 2 {\displaystyle x_{2}} , … from X {\displaystyle X} contains an increasing...