Chains and nested spaces

A chain is a totally ordered poset, and a nested space is a topological space whose lattice of open sets is a chain. That may seem like a curious notion, although you might say that the Scott topology on the real line makes it a nested space—so you know that there at least one natural example of the concept. I will show that nested spaces and chains have very strong topological properties. To start with, I will show you why every chain is a continuous poset. I will then tell you how nested spaces arise from the study of so-called minimal Tand TD topologies, as first explored by R. E. Larson in 1969. And I will conclude with a simple proof of a recent theorem by Mike Mislove. Read the full post.

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