# Examples of Non-Hausdorff Topology [duplicate]

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.

• 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. – Henno Brandsma Sep 13 at 11:15
• This thread from a little over a month ago has quite a few interesting answers: math.stackexchange.com/questions/3778134/… – Tabes Bridges Sep 13 at 11:23
• @Henno Brandsma thank you – Laurence PW Sep 13 at 11:54
• @Tabes Bridges thank you, it seems I didn't search well – 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.
• Thank you for you answer, does the non-Hausdorffness of $\mathcal{L}^p$ play important role somewhere? – Laurence PW Sep 13 at 12:29
• 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. – Michael Heins Sep 13 at 15:45