The Mudcat Café TM
Thread #166522 Message #4108257
Posted By: Rapparee
30-May-21 - 09:24 PM
Thread Name: MOAB - Mother of All BS [annex]
Subject: RE: MOAB - Mother of All BS [annex]
Don't worry about it as paracompact manifolds have all the topological properties of metric spaces. In particular, they are perfectly normal Hausdorff spaces. Manifolds are also commonly required to be second-countable. This is precisely the condition required to ensure that the manifold embeds in some finite-dimensional Euclidean space. However, non-Euclidean Hausdorff spaces do exist (probability approaches certainty) as the existence of manifolds in Hausdorff spaces includes non-Euclidean manifolds.