= Topological manifold
{title2=$M$}
{wiki}
= Manifold
{synonym}
A topological $n$-manifold without boundary is a <topological space> $M$ satisfying three conditions: it is <Hausdorff>, it is <second countable>, and every point $p\in M$ lies in an <open set> $U\subseteq M$ with a <homeomorphism> $\varphi:U\to V$ onto an <open set> $V\subseteq\mathbb R^n$. Here $n\geq0$ is a fixed <dimension of a manifold>, and $\mathbb R^n$ has the usual <topology> of <Euclidean space>. The pair $(U,\varphi)$ is a <coordinate chart>. Unless explicitly qualified, “topological manifold” here means without boundary.<Topology I: smooth manifolds, University of Toronto MAT1300HF, 2012/Week 1 course notes>[Topology I: smooth manifolds, Week 1, Section 1.1, Definition 1.1, page 1]{ref}
The <Hausdorff> condition requires disjoint <open> <neighbourhoods> for distinct points. <Second countability> requires a <countable> <basis of a topology>. Neither follows merely from the local <homeomorphisms>: <non-Hausdorff manifolds> fail the first condition, while an uncountable <disjoint union> of copies of the <real line> fails the second. Indeed, a <basis of a topology> must supply a different nonempty member inside each copy, so it cannot be <countable>.
The <circle> and the <real line> are both one-dimensional <topological manifolds>, with the same local model. Their global <topologies> differ: the <circle> is <compact>, while the <real line> is not. By contrast, the junction of a <topological tripod> already fails the local test. Thus local <coordinate charts> specify what kind of small pieces are allowed; global <topological invariants> distinguish how those pieces fit together.<Introduction to Topological Manifolds, second edition by John M. Lee/Introduction>[Introduction to Topological Manifolds by John M. Lee, second edition, Chapter 1, pages 1–3 on local models and pages 5–6 on topological invariants]{ref}
The usefulness of a <topological manifold> is that local questions about <continuity> can be transported into <Euclidean space>, while <compactness>, <connectedness>, and the <fundamental group> still record global information. For <calculus>, further structure is needed: a <smooth atlas> requires <smooth transition maps>, letting the <chain rule> reconcile <derivatives> computed in different <coordinate charts>. A <topological atlas> by itself only guarantees <homeomorphisms> between coordinates.<Topology I: smooth manifolds, University of Toronto MAT1300HF, 2012/Week 2 course notes>[Topology I: smooth manifolds, Week 2, Section 1.2, equation (11) and Definitions 1.15–1.16, pages 5–6]{ref}
Back to article page
