Topological Completion & Uniformity

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Abstractions about topological and uniform structures, completion, duality, stratification, neighborhood order, and constructions that organize generalized spaces.

16 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Arithmetic progression topologies — Topologies on the integers generated by selected arithmetic progressions as a basis, linking divisibility and congruence structure to topological properties.
• Balanced set — A subset of a real or complex vector space closed under multiplication by every scalar of absolute value at most one.
• Bott cannibalistic class — A K-theory characteristic class measuring how an Adams operation acts on the Thom class of a complex vector bundle or representation.
• Cylinder set — A subset of a Cartesian product determined by restrictions on only finitely many coordinates, forming basic sets for product topology and generators for cylinder sigma-algebras.
• Formal ball — An ordered pair of a metric-space point and a nonnegative radius, ordered so one pair computationally approximates another; generalized formal balls allow negative radii.
• Hausdorff completion — The inverse-limit completion of a filtered group formed from its discrete quotients, with the filtration intersection removed by the canonical map.
• Lawson topology — The common refinement of the Scott topology and lower topology on a poset, combining approximation-sensitive opens with complements of principal upper sets.
• Partition of unity — A locally finite family of nonnegative continuous or smooth functions whose pointwise sum is one, usually with each support subordinate to a member of an open cover.
• Pontryagin duality — A duality for locally compact abelian groups that assigns each group its continuous characters into the circle group and naturally identifies the original group with its double dual.
• Profinite group — A compact totally disconnected Hausdorff topological group expressible as an inverse limit of finite discrete groups.
• Restricted product — The subgroup of a direct product whose coordinates lie in designated compact open subgroups at all but finitely many indices.
• Sorgenfrey plane — The product of two Sorgenfrey lines, a classic separable first-countable space that is not normal and exposes failures of product preservation in topology.
• Specialization preorder — The canonical preorder on points of a topological space in which one point is below another when it belongs to the closure of the other, subject to an explicitly stated orientation convention.
• Stratified space — A topological space decomposed into disjoint manifold-like strata fitted together under frontier and regularity conditions that organize singular behavior by dimension.
• Total subset — A subset of a topological vector space whose linear span is dense in the entire space.
• Uniform property — A property of a uniform space preserved by uniform isomorphisms, including features such as completeness and total boundedness that need not be purely topological.