# Non-Hausdorff Spaces?

## Main Question or Discussion Point

Hi,

Since i will not be asking for assistance in a particular problem, i am posting under "General Math" rather than "Topology". If inappropriate, please move it somewhere else.

As the title suggests, i am interested in reading more about Topological spaces that are NOT Hausdorff, but there seem to be very little on this topic out there, at least compared to what i was able to find about Hausdorff spaces.

So, i was wondering whether you could provide me with some sources, like articles, books etc. that talk exclusively about non-Hausdorff spaces. I have been able to find a few things, but i was wondering whether there is something more in depth.

Thanks!

## Answers and Replies

HallsofIvy
Homework Helper
You probably won't find much. "Non-Hausdorf" spaces are not very interesting- they are used mainly as "counter-examples" to show exactly why Hausforf is needed in so many theorems.

CRGreathouse
Homework Helper
The study of non-Hausdorff spaces properly belongs to set theory, not topology. They just don't have enough nice properties to use topological notions.

Hurkyl
Staff Emeritus
Gold Member
The spaces studied in algebraic geometry are typically non-Hausdorff....

Non-hausdorff topologies occur mostly when you are dealing with problems that demand compactness, but this is incompatible with a strong separation property like Hausdorff's.

Look, for example, at the cofinite and Zarisky topologies; these arise naturally in algebra and algebraic geometry.

In analysis, these topologies also appear when you want to talk about, for example, upper (or lower) semicontinuity of functions; these are the functions that are continuous relative to the topology on $\mathbb R$, generated by the half-open intervals; it is strictly weaker than the standard one and it's not hausdorff. These are important in the modern theories of variational calculus, non-linear analysis and set-valued functions (with applications to game theory and mathematical economics). One note: most textbooks in these fields hide the fact that they are working with a weaker topology by defining upper and lower semicontinuity in terms of upper and lower limits of sequences. But you could take a look at:

Infinite Dimensional Analysis: A Hitchhiker's Guide 3d ed, Aliprantos and Border, Springer, 2006.

Another area where NHT are used is in some of the categorial formulations of logical systems, but Hurkyl could probably tell you much more about these than I. Nevertheless, you may want to take a look at: