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- Hausdorff-Dimension {f} = Hausdorff dimension
- Hausdorff-Raum {m} = Hausdorff space
- Hausdorff'scher Raum {m} = T2 space [Hausdorff space]
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- Because non-Hausdorff manifolds are locally homeomorphic to Euclidean space, they are locally metrizable (but not metrizable) and locally Hausdorff (but not Hausdorff).
- Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom.
- Conversely, every compactly generated Hausdorff space is a quotient of some locally compact Hausdorff space.
- For example, in a non-hausdorff space, it is possible for a sequence to converge to multiple different limits.
- Every Hausdorff space is necessarily sequentially Hausdorff. A sequential space is Hausdorff if and only if it is sequentially Hausdorff.
- Note that the theorem still holds (perhaps vacuously) for "X" an arbitrary Hausdorff space and "Y" a Hausdorff space with countable π-base.
- A variant of Falconer's conjecture states that, for points in the plane, a compact set whose Hausdorff dimension is greater than or equal to one must have a distance set of Hausdorff dimension one.
- Any product of Hausdorff spaces is again a Hausdorff space.
- In mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence.
- Since the only Hausdorff topology on a finite set is the discrete one, a finite Hausdorff topological group must necessarily be discrete.
- This subsection details how every non-Hausdorff TVS [...] can be TVS-embedded onto a dense vector subspace of a complete TVS.
- There exist spaces which are Hausdorff but not Urysohn, and spaces which are Urysohn but not completely Hausdorff or regular Hausdorff. Examples are non trivial; for details see Steen and Seebach.
- However, the compact Hausdorff spaces are "absolutely closed", in the sense that, if you embed a compact Hausdorff space [...] in an arbitrary Hausdorff space [...] then [...] will always be a closed subset of [...]; the "surrounding space" does not matter here.
- The topology [...] need not be Hausdorff but [...] is Hausdorff.
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