Manifold Atlas:Definition of “manifold”

Revision as of 10:29, 16 September 2009

This page defines the term “manifold” as used in the Manifold Atlas.

1 Definition

We assume that all manifolds are of a fixed dimension n. An n-dimensional manifold $This page defines the term “manifold” as used in the Manifold Atlas. <!-- and summarises a few key properties of all manifolds --> == Definition == <wikitex>; We assume that all manifolds are of a fixed dimension n. An '''n-dimensional manifold''' $M$ is a [[Wikipedia:Second_countable|second countable]] [[Wikipedia:Hausdorff_space|Hausdorff space]] for which every point $m \in M$ has a neighbourhood $U_m$ homeomorphic to an open subset of $\Rr^n$ or an open subset of $\Rr^n_+ = \{ x \in \Rr^n | x_1 \geq 0 \}$. The former points are the interior points of $M$. * The '''interior''' of $M$, denoted $\rm{int}(M)$ is the subset of points for which $U_m \subset \Rr^n$. * The '''boundary''' of $M$, written $\partial M$, is the compliemnt of the interior of $M$ * $M$ is called '''closed''' if $M$ is compact and $\partial M$ is empty. === Extra structures === Typically, but not necessarly, the word ``manifold`` will mean $M$ as above with extra structure. The extra structure may or may not be emphasised in notation and vocabulary. * A smooth manifold $(M, \alpha)$ is a manifold $M$ with an equivalence class of </wikitex> == References == {{#RefList:}} [[Category:Theory]] {{Stub}}M$ is a second countable Hausdorff space for which every point $m \in M$ has a neighbourhood $U_m$ homeomorphic to an open subset of $\Rr^n$ or to an open subset of $\Rr^n_+ = \{ x \in \Rr^n | x_1 \geq 0 \}$. The former points are the interior points of $M$.

• The interior of $M$, denoted $\mathrm{int}(M)$, is the subset of points for which $U_m \subset \Rr^n$.
• The boundary of $M$, written $\partial M$, is the compliemnt of the interior of $M$.
• $M$ is called closed if $M$ is compact and $\partial M$ is empty.

Extra structures

Typically, but not necessarly, the word ``manifold`` will mean $M$ as above with extra structure. The extra structure may or may not be emphasised in notation and vocabulary.

• A smooth manifold $(M, \alpha)$ is a manifold $M$ with an equivalence class of

2 References