to 1 as n tends to infinity. Sometimes, the latter definition is modified so that the graph is chosen randomly in some other way, where not all graphs with n vertices have the same probability, and those modified definitions are not always equivalent to the main one. The use of the term "almost all" in graph theory is not standard; the term "asymptotically almost surely" is more commonly used for this concept. Example: Almost all graphs are asymmetric. Almost all graphs have diameter 2. Meaning in topology In topology and especially dynamical systems theory (including applications in economics), "almost all" of a topological space's points can mean "all of the space's points but those in a meagre set". Some use a more limited definition, where a subset only contains almost all of the space's points if it contains some open dense set. Example: Given an irreducible algebraic variety, the properties that hold for almost all points in the variety are exactly the generic properties. This is due to the fact that in an irreducible algebraic variety equipped with the Zariski topology, all nonempty open sets are dense. Meaning in algebra In abstract algebra and mathematical logic, if U is an ultrafilter on a set X, "almost all elements of X" sometimes means "the elements of some element of U". For any partition of X into two disjoint sets, one of them will necessarily contain almost all elements of X. It is possible to think of the elements of a filter on X as containing almost all elements of X, even if it isn't an ultrafilter. Proofs See also Almost Almost everywhere Almost surely References Primary sources Secondary sources Mathematical terminology