| 21. | Recall that a continuum is a nonempty connected compact metric space.
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| 22. | The open balls of a metric space are a unions of open balls.
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| 23. | So, S must be infinite or the metric space is trivial.
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| 24. | In order to prove this theorem Gromov introduced a convergence for metric spaces.
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| 25. | A large class of spaces satisfying the countable first axiom are metric spaces.
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| 26. | A set with a metric is called a metric space.
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| 27. | There are, however, topological spaces that are not metric spaces.
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| 28. | One can generalize the notion of energy distance to probability distributions on metric spaces.
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| 29. | The Baire category theorem says that every complete metric space is a Baire space.
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| 30. | Paracompact manifolds have all the topological properties of metric spaces.
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