| 1. | This construction is dual to the construction of the subspace topology.
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| 2. | Notice that each stalk has the discrete topology as subspace topology.
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| 3. | Generalizing this, we arrive at the definition of the subspace topology.
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| 4. | The subspace topology and product topology constructions are both special cases of initial topologies.
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| 5. | Embedded submanifolds always have the subspace topology.
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| 6. | Every subgroup of a topological group is itself a topological group when given the subspace topology.
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| 7. | It is not necessarily true that the subspace topology is the same as the group topology.
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| 8. | The rational numbers, as a subspace of the real numbers, also carry a subspace topology.
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| 9. | Then the subspace topology on S is defined as the coarsest topology for which \ iota is continuous.
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| 10. | That is, the submanifold topology on " S " is the same as the subspace topology.
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