Definitions
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 adj. Such that any two points have disjoint neighborhoods.
Etymologies
from Wiktionary, Creative Commons Attribution/ShareAlike License
Examples

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

Further, the solution space moduli space is nonHausdorff, which makes things complicated.
Could a Black Hole Fit in Your Computer or In Your Pocket?  Universe Today

(LindenbaumTarski algebras and model theory), set theory (fields of sets), topology (totally disconnected compact Hausdorff spaces), foundations of set theory (Booleanvalued models), measure theory

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.

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

Interestingly the moduli space for general relativity, due to the hyperbolic nature of the connection oneforms, is nonHausdorff.

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

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.

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

There he came across the idea of the HausdorffBesicovitch dimension  the revelation that there were phenomena that existed outside onedimensional space, but in somewhat less than two dimensions.
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