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Any connected 1-manifold non-homeomorphic to admits an orientation reversing map. Thus, is chiral and connected 1-manifolds non-homeomorphic to are amphicheiral. Thus, there are 5 topological types of oriented connected 1-manifolds: Of course, the basic invariants considered above do the job in a more elementary way.

Above by homology we mean homology with compact support. For homology with closed support Borel-Moore homology see the following table: So there are isomorphisms A local coefficient system on a 1-manifold homeomorphic to the circle, may be non-trivial.

The Euler characteristic of is 0. Any closed 1-manifold has Euler characteristic 0. The Euler characteristic of any compact 1-manifold is half the number of its boundary points. When it comes to non-compact spaces, the notion of Euler characteristic becomes ambiguous.

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A non-compact space has many Euler characteristics. The notion of Euler characteristic depends on the properties that one wants to preserve. If one wants to have invariance under homotopy equivalences, then the additivity would be lost.

If one wants to keep additivity, then it is not the alternating sum of Betti numbers. If one wants to keep topological invariance and additivity, thensimilarly. For not necessarily compact 1-manifolds probably most useful is the Euler characteristic which is defined as the alternating sum of the ranks of homology groups with closed support Borel-Moore homology.

It is additive and a topological invariant. The only property which may confuse a person with purely compact experience is that it is not homotopy invariant.

With this Euler characteristic, and.is and in to a was not you i of it the be he his but for are this that by on at they with which she or from had we will have an what been one if would who has her. is and in to a was not you i of it the be he his but for are this that by on at they with which she or from had we will have an what been one if would who has her.

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