Topology

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18 domain-specific abstractions whose origin domain is Topology.

• Cantor set — The compact perfect nowhere-dense subset obtained by repeatedly deleting open middle thirds from a closed interval.
• Completely metrizable space — A topological space whose topology is induced by at least one complete metric, whether or not every compatible metric is complete.
• Cover (topology) — A family of subsets whose union contains a specified set or space, with open covers restricting the members to open subsets.
• 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.
• Erdős space — The subspace of square-summable real sequences whose every coordinate is rational, with the topology inherited from Hilbert space.
• Exhaustion by compact sets — A nested sequence of compact subsets whose interiors successively contain earlier terms and whose union covers the whole topological space.
• Fort space — The one-point compactification of an infinite discrete space, with neighborhoods of the distinguished point having finite complements.
• Hausdorff Space — A topological space in which every two distinct points admit disjoint open neighborhoods, equivalently one whose diagonal is closed and whose convergent nets have unique limits.
• 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.
• Open and closed maps — Maps of topological spaces classified by whether images of every open set or every closed set retain the corresponding property.
• Regular space — A topological space in which every point can be separated from every disjoint closed set by disjoint open neighborhoods.
• Scattered Space — Classify a topological space by requiring every nonempty subspace to expose an isolated point, equivalently that transfinite removal of isolated points eventually exhausts it.
• 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.
• Stratified space — A topological space decomposed into disjoint manifold-like strata fitted together under frontier and regularity conditions that organize singular behavior by dimension.
• Topological property — A property of topological spaces invariant under homeomorphism, depending only on open-set structure rather than coordinates, metric presentation or embedding.
• Totally disconnected space — A topological space whose only connected subspaces are single points and the empty set.