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.
Asked
Active
Viewed 217 times
0
-
1Scott 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 '20 at 11:15
-
1This thread from a little over a month ago has quite a few interesting answers: https://math.stackexchange.com/questions/3778134/naturally-occurring-non-hausdorff-spaces/3778177#3778177 – Tabes Bridges Sep 13 '20 at 11:23
-
@Henno Brandsma thank you – Laurence PW Sep 13 '20 at 11:54
-
@Tabes Bridges thank you, it seems I didn't search well – Laurence PW Sep 13 '20 at 11:54
1 Answers
1
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.
Michael Heins
- 601
-
Thank you for you answer, does the non-Hausdorffness of $\mathcal{L}^p$ play important role somewhere? – Laurence PW Sep 13 '20 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 '20 at 15:45
-