#fibrations

A cylinder is the product of an interval and a circle because it is both an interval of circles and a circle of intervals.

Fibre bundles generalise covering spaces.

Every Seifert-fibered manifold has a geometric structure … of 𝔼³, S³, H²×ℝ, S²×ℝ,S͠L₂, or Nil.

[T]he space of directions in a plane … can be identified with the standard circle S¹ after picking an orientation and a reference direction. Now [picture that] not on the plane, but on a sphere, and specifically, on the surface 𝑿 of the earth. At each point x on this surface, there is a circle Sₓ of directions that one can travel along the sphere from x; the collection S𝑿 ≝ (Sₓ)x∈𝑿 of all such circles is then a circle bundle with base space 𝑿 (known as the circle bundle; it could also be viewed as the sphere bundle, cosphere bundle, or orthonormal frame bundle of 𝑿). The structure group of this bundle is the circle group if one preserves orientation, or the semi-direct product otherwise.