My first thought was: We need a reasonable partition of unity in which the sums are indexed by the natural numbers. This would probably require even normal topologies. The two specific properties (second accountability aka complete separability and Hausdorff) both occur in Urysohn's metrization theorem:littlemathquark said:TL;DR Summary: Why do we need second countable and Hausdorff conditions for manifold definition?
Why do we need second countable and Hausdorff conditions for manifold definition?
However, ...One of the first widely recognized metrization theorems was Urysohn's metrization theorem. This states that every Hausdorff second-countable regular space is metrizable.
... and here we are, back at the partition of unity.It follows that every such space is completely normal as well as paracompact.
Indeed. I was so used to reading Riemannian and even smooth, nice charts, metrics, and spheres that I almost forgot the topological aspect. The two concepts of Urysohn are somehow the minimum we want to have.dextercioby said:"Manifold" is improper usage. One has "topological manifolds" and a very important subset of all of these, the "complex or real differential manifolds".
That is a strange question! Why do you need the other conditions in the definition of a manifold? You can study whatever objects you want, why is a question only you can answer. Some people do look at non Hausdorff manifolds. https://en.wikipedia.org/wiki/Non-Hausdorff_manifoldlittlemathquark said:TL;DR Summary: Why do we need second countable and Hausdorff conditions for manifold definition?
Why do we need second countable and Hausdorff conditions for manifold definition?