Topological Spaces & Compactness

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Abstractions about topological spaces characterized by compactness, countability, metrizability, connectedness, separation, and mapping properties. They include special constructed spaces, embeddings, filters, isolated and adherent points, uniformization, and behavior at infinity.

26 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.
• 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.
• 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.
• Development (topology) — A countable sequence of open covers whose stars at each point form a neighborhood base, characterizing developable spaces.
• Discrete space — A topological space in which every subset is open, equivalently every point is isolated and the topology is the full power set.
• 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.
• 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.
• Fort space — The one-point compactification of an infinite discrete space, with neighborhoods of the distinguished point having finite complements.
• H-closed space — A Hausdorff topological space that is closed in every Hausdorff space in which it embeds as a subspace.
• Isolated point — A point of a subset having a neighborhood that contains no other point of that subset.
• 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.
• 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.
• 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.
• Regular space — A topological space in which every point can be separated from every disjoint closed set by disjoint open neighborhoods.
• 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.
• Simply connected at infinity — A noncompact space property requiring sufficiently remote loops to contract outside any prescribed compact core.
• 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.
• 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.
• Γ-space — A topological space in which every open omega-cover contains a gamma-cover whose members contain each point all but finitely often.