When I learned real analysis and topology courses, most of the spaces are Hausdorff. I only know Zariski topology and étale topology for non-Hausdorff ones that play important roles in algebraic geometry. Are there more important examples of non-Hausdorff topology? Any answers and references are welcome.

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    $\begingroup$ Scott topologies (see Wikipedia) play a role in theoretical computer science and other fields close to logic. They are often non-Hausdorff too. Also the digital line (and its square) is a commonly cited example to model pixels on screen. $\endgroup$ – Henno Brandsma Sep 13 at 11:15
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    $\begingroup$ This thread from a little over a month ago has quite a few interesting answers: math.stackexchange.com/questions/3778134/… $\endgroup$ – Tabes Bridges Sep 13 at 11:23
  • $\begingroup$ @Henno Brandsma thank you $\endgroup$ – Laurence PW Sep 13 at 11:54
  • $\begingroup$ @Tabes Bridges thank you, it seems I didn't search well $\endgroup$ – Laurence PW Sep 13 at 11:54

The $\mathcal{L}^p$ function spaces are an example one usually does not think of as a non Hausdorff topology. In practice one jumps between $\mathcal{L}^p$ and the corresponding Hausdorffization $$L^p = \mathcal{L}^p \big/ \{0\}^{\textrm{cl}},$$ as working with representatives, i.e. functions instead of equivalence classes, is more intuitive in many places.

Another notable class of examples arises in Oid-Geometry: the total space of all arrows in a Lie groupoid is non Hausdorff in many important examples such as the Monodromy and the Holonomy groupoids.

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  • $\begingroup$ Thank you for you answer, does the non-Hausdorffness of $\mathcal{L}^p$ play important role somewhere? $\endgroup$ – Laurence PW Sep 13 at 12:29
  • $\begingroup$ Only in the sense that one immediately sets out to fix that by passing to $L^p$ instead. Most of the interesting theorems only hold for the equivalence classes. $\endgroup$ – Michael Heins Sep 13 at 15:45
  • $\begingroup$ okay, thank you $\endgroup$ – Laurence PW Sep 14 at 4:58

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