Separation axiom

A condition on a topological space describing how points or closed sets can be distinguished by open sets. The axioms form a hierarchy, including the familiar Hausdorff and regular conditions.

Topological space: A set equipped with a collection of subsets designated as open and satisfying three axioms. Separation axioms impose additional conditions on this underlying structure.
Kolmogorov space: A topological space in which any two distinct points have different neighborhoods. This weakest common condition separates points by neighborhood behavior, not necessarily by disjoint sets.
T1 space: A topological space in which every singleton is closed. The T1 condition is stronger than mere distinguishability and weaker than Hausdorffness.
Compact space: A topological space in which every open cover has a finite subcover. Compactness combines with Hausdorffness to yield strong uniqueness and closedness results.
Open set: A member of a topology, defined by the axioms that unions and finite intersections of such sets remain in the topology. The axioms specify which open neighborhoods must separate points or closed sets.
Hausdorff space: A topological space in which every pair of distinct points has disjoint open neighborhoods. The Hausdorff condition gives a direct, pairwise separation of distinct points.
Tychonoff space: A completely regular Hausdorff topological space. This widely used combination clarifies which conventions accompany the name Tychonoff.
One-point compactification: A construction adjoining one point to a noncompact locally compact Hausdorff space to make it compact. The Hausdorff condition helps ensure the added point produces a useful compact space.
Closed set: A subset whose complement is open. Several separation conditions ask whether closed sets can be separated from points or other closed sets.
Regular space: A topological space in which a point and a disjoint closed set can be separated by disjoint open neighborhoods. Regularity extends separation from point pairs to a point and a closed set.

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