Topological manifolds with and without boundary

Fix an integer $n\ge 0$. For $n\ge 1$, let ${\mathbb{R}}_{+}^n=\{(t_1,\dots ,t_n):t_n\ge 0\}$, with the subspace topology of Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace, and let its model boundary be $\{t_n=0\}$. For $n=0$ use the one-point space ${\mathbb{R}}^0$ and stipulate that its model boundary is empty.

An $n$-dimensional topological manifold with boundary is a Hausdorff space $M$, in the sense of Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not, which has a countable basis of open sets and for which every point has an open neighborhood $U$ with a homeomorphism $h:U\to V$, where $V$ is open in ${\mathbb{R}}_{+}^n$. Here countable means at most countable, including finite and empty, as in Finite, countably infinite, countable, uncountable, and basis means the open-set basis of Basis and subbasis for a topology, and the topology generated by a family of sets. The pair $(U,h)$ is a chart. Homeomorphism means a bijection continuous in both directions.

Define $\mathrm{\partial }M$ to be the set of points sent into the model boundary by at least one such chart, and define $\mathrm{int}M=M\setminus \mathrm{\partial }M$. This is an unambiguous subset: the quantifier ranges over all charts, rather than over a chosen atlas. The subsequent local-homology theorem proves the stronger assertion that every chart agrees about boundary membership, and proves that this convention agrees with the locally Euclidean definition of a manifold without boundary. That assertion is not assumed in the definition of the subset.

A manifold without boundary, or boundaryless manifold, is one for which $\mathrm{\partial }M=\mathrm{\varnothing }$. Such a manifold is locally Euclidean directly: at any point choose a chart, whose image point lies strictly above the hyperplane, then restrict to a Euclidean ball lying in its image and above that hyperplane. For $n=0$ each chart domain is a singleton; hence the space is discrete and its boundary is empty by convention.

Connectedness and nonemptiness are not required. The empty space satisfies the definition for every specified $n$ and has empty boundary; its dimension label cannot be recovered from its underlying space. A point is a zero-dimensional example. All chart choices in this definition are individual existential hypotheses, not a simultaneous choice of charts or an application of AC.