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 to 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 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 '''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 | + | * 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. | ||
Revision as of 11:44, 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 is a second countable Hausdorff space for which every point has a neighbourhood homeomorphic to an open subset of or to an open subset of . The former points are the interior points 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.
Extra structures
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. |