The more typical definition is that a manifold is a space that is locally like $\mathbb {R}^n$. I dislike it philosophically, but it's easier to formalize than M1-M3 above.

Mar 20, 2015 · 37 Now we always encounter definition of a manifold from a mathematical point of view where it is a topological space along with a family of open sets that covers it and the …

From Wikipedia: The broadest common definition of manifold is a topological space locally homeomorphic to a topological vector space over the reals. A topological manifold is a …

Jun 15, 2019 · A "closed manifold" is a topological space that has the following properties: it is a manifold [locally Euclidean, second countable, Hausdorff topological space] that is additionally …

Apr 8, 2016 · A manifold, in Riemann's words, is a continuous transition of an instance I know a manifold is something glued by locally homeomorphic to Euclidean parts, but from that I can't …

Jun 29, 2022 · 75 I was studying some hyperbolic geometry previously and realised that I needed to understand things in a more general setting in terms of a "manifold" which I don't yet know …

Oct 15, 2020 · All the neighborhoods in definition 1 are the open cover from definition 2, and an open cover like in definition 2 contains neighborhoods of all points, so it also fulfills definition 1.

Aug 8, 2015 · The definition of a smooth manifold shouldn't be complicated since it's a fairly intuitive object. Practically speaking if you wanna see whether a topological space is a …

NOT A DUPLICATE: Homeomorphism in the definition of a manifold for example is slightly different. A manifold according to Wikipedia, a book by Spivak, and serveral other books has …

Nov 3, 2025 · So my second question is what is actually going on here? how is a structure over a topological manifold defined? what equivalence relation is assumed? Are any of these …