Convention: manifolds are smooth, Hausdorff, and second countable

Convention

Unless explicitly stated otherwise, the word manifold means a smooth manifold $M$ satisfying:

1. Smoothness: $M$ is a $C^\infty$ manifold (all atlases, transition maps, and structure maps are smooth).
2. Hausdorff: For any distinct points $p\neq q$ in $M$, there exist disjoint open sets separating them.
3. Second countable: The topology of $M$ has a countable basis.

This convention ensures standard foundational results used throughout differential geometry, including existence of partitions of unity, paracompactness, and well-behaved constructions of vector bundles and principal bundles .

Examples

1. Why Hausdorff matters. The “line with two origins” is a classical example of a non-Hausdorff topological manifold; many standard arguments in analysis and geometry (e.g., uniqueness of limits) fail there.

2. Why second countable matters. A long line is Hausdorff and locally Euclidean but not second countable; it is not paracompact, and partitions of unity can fail to exist.

3. Typical spaces covered by the convention. Open subsets of $\mathbb R^n$, tori $T^n$, spheres $S^n$, and smooth quotients arising from free proper group actions (see quotient manifold construction ) all satisfy these hypotheses.