Hausdorff space

A topological space in which every pair of distinct points has disjoint open neighborhoods. This separation condition is also called the T₂ axiom.

Topological space: A set equipped with a collection of subsets, called open sets, satisfying axioms for unions and finite intersections. Hausdorff separation is an additional condition imposed on a topological space.
Uniqueness of limits: The property that a convergent sequence or net cannot have two distinct limits. Disjoint neighborhoods prevent a sequence or net from eventually approaching two different points.
Sierpiński space: The two-point topological space with open sets ∅, {1}, and {0,1}. Its distinct points cannot be separated by disjoint open neighborhoods.
Hausdorff compactification: A compact Hausdorff space containing a given space as a dense subspace. Compactifications often impose Hausdorffness to obtain well-behaved compact ambient spaces.
Felix Hausdorff: A German mathematician whose 1914 book Grundzüge der Mengenlehre helped establish modern set-theoretic topology. The separation axiom bears his name through his foundational work in topology.
Open set: A member of a topology, whose arbitrary unions and finite intersections remain in that topology. The disjoint neighborhoods in the Hausdorff condition must be open sets.
Closed set: A subset whose complement is open; equivalently, a set containing all its limit points. Every singleton is closed in a Hausdorff space.
Kolmogorov space: A topological space in which any two distinct points have different neighborhoods. This weaker T₀ condition distinguishes points without requiring disjoint neighborhoods.
Stone–Čech compactification: The largest compact Hausdorff compactification of a completely regular topological space. Its defining target combines compactness with Hausdorff separation.
Grundzüge der Mengenlehre: Felix Hausdorff’s 1914 foundational book on set theory and topology. It gave a systematic framework for the separation ideas now associated with Hausdorff spaces.

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