Manifold conventions and the role of second countability

This library adopts the standard finite-dimensional convention that a topological manifold is Hausdorff, second countable, and locally Euclidean (Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces). The third condition gives the local models; the first two are global control assumptions, and they are used rather than merely recorded. In particular, under the explicit choice hypotheses in Topological manifolds are metrizable and paracompact, the global consequences proved on this page, such as metrizability and paracompactness, use the second-countability part of the definition and do not hold for arbitrary Hausdorff locally Euclidean spaces. The convention supplies the topological hypotheses; the cited theorem separately records ${\mathrm{A}\mathrm{C}}_{\omega }$ for its Lindelof step and the Axiom of Choice for its metrization step.

The long line (The closed long ray ${\omega }_1\times [0,1)$ under the lexicographic order, and the long line, with the order topology) is the standard warning. It is built from locally interval-like blocks, so it looks locally like a one-dimensional manifold, but it fails the countability convention and is therefore excluded.

Different texts make different choices here. Some authors call every Hausdorff locally Euclidean space a manifold and add countability only when they need it; this library does not. The reason is structural rather than terminological: the next page uses partitions of unity, and the topological hypotheses needed there are already built into the present convention.