Definitions

from Wiktionary, Creative Commons Attribution/Share-Alike License

  • adj. Such that any two points have disjoint neighborhoods.

Etymologies

From the eponym Felix Hausdorff (1868–1942). (Wiktionary)

Examples

  • Pure math research into the Hausdorff metric geometry proved no two sets have exactly 19 elements in any given location between them.

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  • Further, the solution space moduli space is non-Hausdorff, which makes things complicated.

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  • (Lindenbaum-Tarski algebras and model theory), set theory (fields of sets), topology (totally disconnected compact Hausdorff spaces), foundations of set theory (Boolean-valued models), measure theory

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  • The foundational nature of renormalization is encoded in the renormalization group equation, Wilson and Polchinski, which has a fluid mechanics analogue and describes a Hausdorff dimension for the scalability of the theory.

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  • As for fractals, there is a Hausdorff dimension with the renormalization group of running parameters.

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  • Interestingly the moduli space for general relativity, due to the hyperbolic nature of the connection one-forms, is non-Hausdorff.

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  • Building on the work of Hausdorff, Banach and Tarski derive from

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  • In one direction, an arbitrary Boolean algebra yields a topological space, and in the other direction, from a (compact Hausdorff and totally disconnected) topological space, one obtains a Boolean algebra.

    Category Theory

  • Hausdorff paradox: There exists a countable subset C of the sphere S such that SC is equidecomposable with two copies of itself.

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  • There he came across the idea of the Hausdorff-Besicovitch dimension - the revelation that there were phenomena that existed outside one-dimensional space, but in somewhat less than two dimensions.

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