Filtertopologie

Filtertopologie, literally “filter topology,” is a way of describing a topological space by using filters instead of open sets as the primary data. For each point x in a set X, one assigns a neighborhood filter N_x, a filter of subsets of X that represent the neighborhoods of x. A topology is then equivalent to a compatible family of such filters, and convergence and continuity can be described directly in terms of these filters.

Construction and equivalence: Suppose for every x ∈ X there is a filter N_x on X consisting of subsets that contain x, and these filters satisfy the usual neighborhood axioms. The open sets of the corresponding topology are exactly those subsets U ⊆ X for which U ∈ N_x for every x ∈ U. Conversely, given a topology on X, one obtains a neighborhood filter N_x at each x by taking as members those sets that contain an open neighborhood of x. Thus, a topology and its system of neighborhood filters determine each other; many topological properties and notions (continuity, convergence) can be formulated purely in terms of these filters.

Examples and relationships: In a discrete topology, N_x is the full filter of all subsets containing x; in the indiscrete (trivial) topology, N_x contains only the whole space X. The filter-topology framework is especially natural for describing convergence: a filter F on X converges to x precisely when N_x ⊆ F. The approach also connects with constructions in which ultrafilters and filter convergence play central roles, such as the Stone–Čech compactification.

See also: Topology, Filter (mathematics), Convergence (filters), Ultrafilter, Stone–Čech compactification.