Manifold Atlas:Definition of “manifold”
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+ | == Introduction == | ||
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This page defines the term “manifold” as used in the Manifold Atlas. | This page defines the term “manifold” as used in the Manifold Atlas. | ||
− | <!-- and summarises a few key properties of all manifolds --> | + | <!-- and summarises a few key properties of all manifolds --> |
+ | We assume that all manifolds are of a fixed dimension n. | ||
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+ | {{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 $\Rr^n_+ := \{ x \in \Rr^n | x_1 \geq 0 \}$. | ||
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* The '''interior''' of $M$, denoted $\mathrm{int}(M)$, is the subset of points for which $U_m \subset \Rr^n$. | * 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 $M$. | * 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. | * $M$ is called '''closed''' if $M$ is compact and $\partial M$ is empty. | ||
+ | {{endthm}} | ||
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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. | 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 | * A smooth manifold $(M, \alpha)$ is a manifold $M$ with an equivalence class of |
Revision as of 19:58, 16 September 2009
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 is a second countable Hausdorff space for which every point has a neighbourhood homeomorphic to an open subset of .
- The interior of , denoted , is the subset of points for which .
- The boundary of , written , is the complement of the interior of .
- is called closed if is compact and is empty.
Typically, but not necessarly, the word “manifold” will mean as above with extra structure. The extra structure may or may not be emphasised in notation and vocabulary.
- A smooth manifold is a manifold with an equivalence class of
2 References
This page has not been refereed. The information given here might be incomplete or provisional. |