Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces

Let $n\in \mathbb{N}$. For $n\ge 1$, put ${\mathbb{R}}^n:={\prod }_{k<n}\mathbb{R}$ with its usual topology; for $n=0$ put ${\mathbb{R}}^0:=\{0\}$, the one-point space. A topological $n$-manifold without boundary (or briefly an $n$-manifold) is a topological space $M$ satisfying:

1. $M$ is Hausdorff (Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not);
2. $M$ is second countable (Second countability: an at most countable basis for the topology);
3. $M$ is locally Euclidean of dimension $n$: every $p\in M$ has an open neighbourhood $U\subseteq M$ homeomorphic to an open subset of ${\mathbb{R}}^n$ (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological).

The empty space satisfies all three conditions vacuously, so $M=\mathrm{\varnothing }$ is an $n$-manifold for every $n$; this degenerate instance is kept, and statements about nonempty manifolds name the hypothesis. In dimension zero, condition 3 forces the one-point neighbourhoods of points to be open singletons, so a $0$-manifold is exactly a discrete second-countable space with at most countably many points.

• Second countability is part of the definition. It is exactly the axiom that the long line fails, and it is what later turns the local hypotheses into global ones: countable chart bases, $\sigma$-compactness, metrizability and paracompactness all flow from it. The convention split is recorded in Manifold conventions and the role of second countability.

• No boundary is defined here. A manifold with boundary replaces the local models by open subsets of the closed upper half-space; that is a strictly later construction and no statement on this page silently permits it.