1-manifolds
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− | According to general definition of manifold, a manifold of dimension 1 is a topological space which is second countable (i.e., has a countable base), satisfies the Hausdorff axiom (any two different point in it have disjoint neighborhoods) and each point of which has a neighborhood homeomorphic either to line $\ | + | According to general definition of manifold, a manifold of dimension 1 is a topological space which is second countable (i.e., has a countable base), satisfies the Hausdorff axiom (any two different point in it have disjoint neighborhoods) and each point of which has a neighborhood homeomorphic either to line $\Rr $ or half-line $\Rr_+=\{x\in\mathbb R\mid x\ge0\}$. |
Manifolds of dimension 1 are called ''curves'', but this name may lead to a confusion, because many mathematical objects share it. Even in the context of topology, the term ''curve'' may mean not only a manifold of dimension 1 with an additional structure, but, for instance, an immersion | Manifolds of dimension 1 are called ''curves'', but this name may lead to a confusion, because many mathematical objects share it. Even in the context of topology, the term ''curve'' may mean not only a manifold of dimension 1 with an additional structure, but, for instance, an immersion |
Revision as of 19:39, 30 September 2009
An earlier version of this page was published in the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 08:32, 18 July 2013 and the changes since publication. |
Contents |
1 Introduction
According to general definition of manifold, a manifold of dimension 1 is a topological space which is second countable (i.e., has a countable base), satisfies the Hausdorff axiom (any two different point in it have disjoint neighborhoods) and each point of which has a neighborhood homeomorphic either to line or half-line .
Manifolds of dimension 1 are called curves, but this name may lead to a confusion, because many mathematical objects share it. Even in the context of topology, the term curve may mean not only a manifold of dimension 1 with an additional structure, but, for instance, an immersion of a smooth manifold of dimension 1 to Euclidean space . To be on the safe side, we use an unambiguous term manifold of dimension 1 or 1-manifold.
2 Construction and examples
Real line , half-line , circle , closed interval .
3 Invariants
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4 Classification/Characterization
Any 1-manifold is homeomorphic to disjoint sum of its connected components. The connected components of a 1-manifold are 1-manifolds.
Two manifolds are homeomorphic iff there exists a one-to-one correspondence between their components such that the corresponding components are homeomorphic.
Any connected 1-manifold is homeomorphic either to real line , half-line , circle , closed interval .
Any connected closed 1-manifold is homeomorphic to the circle .
Any connected compact 1-manifold with non-empty boundary is homeomorphic to the interval .
Any connected non-compact 1-manifold without boundary is homeomorphic to the line .
Any connected non-compact 1-manifold with non-empty boundary is homeomorphic to the half-line .
Thus we see that for connected 1-manifolds two invariants, compactness and presence of boundary, taking each two values form a complete system of topological invariants.
The classification theorems stated above can be deduced from the following simple Lemma: a Hausdorff topological space which can be covered by two open subsets which are homeomorphic to is homeomorphic either to or .
5 Further discussion
The results above solve topological classification problem for 1-manifolds in the most effective way that one can wish. Surprisingly, many textbooks manage not to mention it.
6 References
This page has not been refereed. The information given here might be incomplete or provisional. |