Point-Set Topology Structures

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Abstractions about point-set topology, covering how open sets, covers, compactness, and separation axioms define topological spaces, including special examples like the Cantor set and Sorgenfrey plane, and topological vector-space structures like Fréchet and nuclear spaces.

51 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.

• Adherent point — A point every neighborhood of which intersects a selected subset, equivalently a member of that subset's closure.
• 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.
• Cantor set — The compact perfect nowhere-dense subset obtained by repeatedly deleting open middle thirds from a closed interval.
• Compact embedding — An embedding whose inclusion map is compact, so bounded sequences in the source possess subsequences converging in the target; in topology, related notation can instead mean compact containment.
• Completely metrizable space — A topological space whose topology is induced by at least one complete metric, whether or not every compatible metric is complete.
• Continuous function — A function that preserves arbitrarily local closeness: inverse images of open sets are open, equivalently limits can pass through the function under suitable structures.
• Conway criterion — A sufficient boundary-symmetry test guaranteeing that a topological-disk prototile can tile the plane by translations and half-turns.
• Core-compact space — A topological space whose lattice of open sets is a continuous poset, equivalently an exponentiable object in the category of topological spaces.
• Cover (topology) — A family of subsets whose union contains a specified set or space, with open covers restricting the members to open subsets.
• Development (topology) — A countable sequence of open covers whose stars at each point form a neighborhood base, characterizing developable spaces.
• Discontinuous linear map — A linear transformation between topological vector spaces that fails the continuity or boundedness condition imposed by their topologies.
• Discrete space — A topological space in which every subset is open, equivalently every point is isolated and the topology is the full power set.
• Dispersion point — Identify a point whose removal turns a connected topological space into a space with no nontrivial connected component, concentrating the original connectedness at one indispensable point.
• Door space — A topological space in which every subset is open, closed or both.
• Exhaustion by compact sets — A nested sequence of compact subsets whose interiors successively contain earlier terms and whose union covers the whole topological space.
• Extremally disconnected space — A topological space in which the closure of every open set is open.
• Filters in topology — A set-family formalism that characterizes convergence, continuity, closure, compactness, and limits in arbitrary topological spaces without relying on sequences.
• First-countable space — A topological space in which every point has a countable neighborhood basis.
• Fréchet space — A complete metrizable locally convex topological vector space, often described by a countable separating family of seminorms and broad enough to include many function spaces that have no single adequate norm.
• H-closed space — A Hausdorff topological space that is closed in every Hausdorff space in which it embeds as a subspace.
• Hypertopology — A topology placed on a hyperspace of subsets, commonly the nonempty closed subsets of a topological space, so sets themselves become continuously varying points.
• Isolated point — A point of a subset having a neighborhood that contains no other point of that subset.
• 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.
• Lebesgue covering dimension — The least integer n such that every open cover of a topological space has an open refinement of order at most n+1.
• Local boundedness — A property requiring a function, family or operator to remain bounded on some neighborhood of every point, without requiring one global bound over the whole domain.
• Locally Hausdorff space — A topological space in which every point has a neighborhood that is Hausdorff in its subspace topology, allowing locally unique limits while global point separation can still fail.
• Menger space — A topological space in which, from every sequence of open covers, finitely many sets can be selected from each cover so that all selected sets together still cover the space.
• Metrizable space — A topological space whose open sets are exactly those generated by some metric on its underlying set.
• Moore space (topology) — A regular Hausdorff topological space possessing a countable development of open covers that locally refines every neighborhood.
• Normal space — A topological space in which every pair of disjoint closed sets can be enclosed in disjoint open neighborhoods, with Hausdorffness required separately for the T4 convention.
• Nuclear space — A locally convex topological vector space whose connecting maps between suitable seminorm completions are nuclear, giving strong finite-dimensional-like compactness and tensor properties.
• Open and closed maps — Maps of topological spaces classified by whether images of every open set or every closed set retain the corresponding property.
• Overlapping interval topology — A topology on the interval minus-one to one generated by left and right half-open intervals whose overlap produces a standard counterexample with distinctive separation properties.
• Parovicenko space — A compact Hausdorff space of continuum weight satisfying characteristic separation and interior conditions modeled on the Stone–Čech remainder of the integers.
• 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.
• Regular space — A topological space in which every point can be separated from every disjoint closed set by disjoint open neighborhoods.
• Restricted product — The subgroup of a direct product whose coordinates lie in designated compact open subgroups at all but finitely many indices.
• Saturated set (intersection of open sets) — A subset of a topological space equal to the intersection of all open sets containing it, equivalently an upper set for the specialization preorder.
• Semiregular space — A topological space whose regular open sets form a base for its topology.
• Sequentially compact space — A topological space in which every sequence has a subsequence converging to a point of the space, coinciding with compactness in metric spaces but not in general.
• 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.
• Supercompact space — A topological space possessing a subbase for which every subbasic open cover has a subcover of at most two members, a strong cover property implying compactness and preserved under products.
• Topological homomorphism — A continuous linear map between topological vector spaces that induces a topological isomorphism from the quotient by its kernel onto its image.
• Total subset — A subset of a topological vector space whose linear span is dense in the entire space.
• Totally disconnected space — A topological space whose only connected subspaces are single points and the empty set.
• Uniformizable space — A topological space whose topology is induced by at least one uniform structure, equivalently a completely regular space under the stated separation convention.
• Webbed space — A topological vector space equipped with a web structure that supports generalized closed-graph and open-mapping theorems.
• Γ-space — A topological space in which every open omega-cover contains a gamma-cover whose members contain each point all but finitely often.