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What are the most interesting topological spaces that satisfy the weakest separation axioms?

When I say interesting I mean in the sense that is beyond merely underestaning topological spaces better. I'm looking more for applications in different fields that use such spaces.

Clearly metrizable spaces have huge relevance. For normal but not metric spaces I can only think of order topologies of large ordinals.

According to Munkres, it seems that it will be hard to find good examples for non-Hausdorff spaces, though I think he is impliying that there are good examples for spaces that do satisfy Hausdorff but not more.

So - I'm curious to hear of different proofs and applications in various fields that use topological space which have a weak separation - the weaker the better.

Thanks!

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  • $\begingroup$ Separability is essential in that it implies the uniqueness of a limit for a convergent sequence, key point of the analysis. By the way, for French a compact it is separated by definition what is not the case for the American (for example the trivial topology is compact in USA but not in France). $\endgroup$ – Piquito Jul 15 '15 at 21:12
  • $\begingroup$ One example I thought of was the Sorgenfrey plane. The Sorgenfrey line itself is helpful when looking at lower limits. I think you can think of a "lower and increasing" limit when thinking of the Sorgenfrey plane (in a way giving constraints on the direction of the limits in more dimensions). At least the Sorgenfrey plane is not normal, even though still quite separable. $\endgroup$ – mdot Jul 16 '15 at 8:01

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