Manifold Atlas:Definition of “manifold”

1 Introduction

This page defines the term “manifold” as used in the Manifold Atlas. We assume that all manifolds are of a fixed dimension n.

Definition 1.1. An n-dimensional topological manifold $M$$== Introduction == <wikitex>; This page defines the term “manifold” as used in the Manifold Atlas. <!-- and summarises a few key properties of all manifolds --> We assume that all manifolds are of a fixed dimension n. {{beginthm|Definition|}} An '''n-dimensional topological manifold''' $M$ is a [[Wikipedia:Second_countable|second countable]] [[Wikipedia:Hausdorff_space|Hausdorff space]] for which every point $m \in M$ has a neighbourhood $U_m$ homeomorphic to an open subset of $\Rr^n_+ := \{ x \in \Rr^n | x_1 \geq 0 \}$. * The '''interior''' of $M$, denoted $\mathrm{int}(M)$, is the subset of points for which $U_m \subset \Rr^n$. * The '''boundary''' of $M$, written $\partial M$, is the complement of the interior of $M$. * $M$ is called '''closed''' if $M$ is compact and $\partial M$ is empty. {{endthm}} Typically, but not necessarly, the word “manifold” will mean $M$ as above with extra structure. The extra structure may or may not be emphasised in notation and vocabulary. * A smooth manifold $(M, \alpha)$ is a manifold $M$ with an equivalence class of </wikitex> == References == {{#RefList:}} [[Category:Theory]] {{Stub}}M$ is a second countable Hausdorff space for which every point $m \in M$$m \in M$ has a neighbourhood $U_m$$U_m$ homeomorphic to an open subset of $\Rr^n_+ := \{ x \in \Rr^n | x_1 \geq 0 \}$$\Rr^n_+ := \{ x \in \Rr^n | x_1 \geq 0 \}$.

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

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

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

2 References

This page has not been refereed. The information given here might be incomplete or provisional.