| 1. | Every manifold has a natural topology since it is locally Euclidean.
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| 2. | The property of being locally Euclidean is preserved by local homeomorphisms.
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| 3. | This much is a fragment of a typical locally Euclidean topological group.
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| 4. | In particular, being locally Euclidean is a topological property.
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| 5. | It is true, however, that every locally Euclidean space is T 1.
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| 6. | A "'topological manifold "'is a locally Euclidean Hausdorff space.
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| 7. | An example of a non-Hausdorff locally Euclidean space is the line with two origins.
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| 8. | The definition I follow is the " locally Euclidean " one so I allow the long line to be a manifold.
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| 9. | In the previous section, a surface is defined as a topological space with certain properties, namely Hausdorff and locally Euclidean.
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| 10. | There are also topological manifolds, i . e ., locally Euclidean spaces, which possess no differentiable structures at all.
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