| 1. | Then, by continuity, the inverse image of that is an open set.
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| 2. | By invariance of domain, the map carries open sets onto open sets.
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| 3. | Note that the neighbourhood V need not be an open set itself.
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| 4. | As elements of the topology, cylinder sets are by definition open sets.
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| 5. | The weak topology is generated by the following basis of open sets:
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| 6. | Every open set is a countable union of a collection of these.
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| 7. | An arbitrary union of open sets in a topological space is open.
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| 8. | That is, the intersection of almost open sets is again almost open.
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| 9. | By invariance of domain, the map carries open sets onto open sets.
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| 10. | In fact, its open sets are even inaccessible by " any " suprema.
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