(mathematics) A field that is also a topological space in which addition and multiplication are both continuous.

plural of Grothendieck topology

(category theory) A structure on a category which makes the category's objects act like the open sets of a topological space.

(algebraic geometry) Originally, a topology applicable to algebraic varieties, such that the closed sets are the variety's algebraic subvarieties; later, a generalisation in which the topological space is the set of prime ideals of a commutative ring and is called the spectrum of the ring.

(topology) subspace topology

(topology) The topology of the real number system generated by a basis which consists of all open balls (in the real number system), which are defined in terms of the one-dimensional Euclidean metric.

(topology) The topology of a subset S of a topological space X which is obtained by considering any subset of S to be an open set if it corresponds to the intersection of S with some open set of X.

(topology) The topology of the Cartesian product of two or more topological spaces which is generated from a basis whose elements are Cartesian products of open subsets of the original spaces, such that for each such Cartesian product only a finite quantity of such subsets are proper.

(graph theory, computer science) An ordering of the vertices of a directed graph such that if an edge goes from vertex u to vertex v then u precedes v in the ordering.

(topology) Given a set X, a family of topological spaces (M_i,T_i)_(i∈I) and a family of functions f_i:X→M_i, the initial topology is the coarsest topology on X such that all f_i are continuous.