Non-Hausdorff spaces by John L. Bell.

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In summary, the conversation discusses the topic of non-Hausdorff topological spaces and their relevance in mathematics. The speaker mentions that these spaces are not very interesting and are mainly used as counter-examples in theorems. They suggest that the study of non-Hausdorff spaces belongs to set theory rather than topology and provide some examples of where these spaces are used, such as in algebraic geometry and analysis. They also recommend some sources for further reading on the topic.
  • #1

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.

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  • #2
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.
  • #3
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.
  • #4
The spaces studied in algebraic geometry are typically non-Hausdorff...
  • #5
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 [itex]\mathbb R[/itex], 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:"
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1. What are Non-Hausdorff Spaces?

Non-Hausdorff Spaces are topological spaces in which the Hausdorff property, also known as the separation axiom, does not hold. This means that there exist points in the space that cannot be separated by disjoint open sets.

2. What is the significance of the Hausdorff property in topological spaces?

The Hausdorff property is an important separation axiom in topology. It ensures that points in a topological space can be distinguished from each other by disjoint open sets, which is useful for many applications in mathematics and physics.

3. What are some examples of Non-Hausdorff Spaces?

Examples of Non-Hausdorff Spaces include the Zariski topology, the lower limit topology, and the Sorgenfrey line. These spaces have interesting properties and are often used in algebraic geometry and analysis.

4. How do Non-Hausdorff Spaces differ from Hausdorff Spaces?

The main difference between Non-Hausdorff Spaces and Hausdorff Spaces is in the separation axiom. In Hausdorff Spaces, any two distinct points can be separated by disjoint open sets, while in Non-Hausdorff Spaces, there exist points that cannot be separated by disjoint open sets.

5. Can Non-Hausdorff Spaces still be useful in mathematics?

Yes, Non-Hausdorff Spaces have many interesting applications in mathematics, especially in algebraic geometry and analysis. They also help to generalize and better understand topological concepts and properties in different contexts.

Suggested for: Non-Hausdorff spaces by John L. Bell.