Manifold Atlas:Definition of “manifold”
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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 $M$. | 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 $ | + | * 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 $M$ | * 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. | * $M$ is called '''closed''' if $M$ is compact and $\partial M$ is empty. |
Revision as of 10:24, 16 September 2009
This page defines the term “manifold” as used in the Manifold Atlas.
1 Definition
Tex syntax erroris a second countable Hausdorff space for which every point has a neighbourhood homeomorphic to an open subset of or an open subset of . The former points are the interior points of
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- The interior of
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, denotedTex syntax error
is the subset of points for which . - The boundary of
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, written , is the compliemnt of the interior ofTex syntax error
-
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is called closed ifTex syntax error
is compact and is empty.
Extra structures
Typically, but not necessarly, the word ``manifold`` will meanTex syntax erroras above with extra structure. The extra structure may or may not be emphasised in notation and vocabulary.
- A smooth manifold is a manifold
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with an equivalence class of
2 References
This page has not been refereed. The information given here might be incomplete or provisional. |