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2. If a collection of subsets of a topological space X is locally discrete, it must satisfy the property that each point of the space belongs to at most one element of the collection. This means that only collections of pairwise disjoint sets can be locally discrete.
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collection of sets is necessarily countably locally discrete. Therefore, if X is a metrizable space with a
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is said to be countably locally discrete, if it is the countable union of locally discrete collections.
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cannot have a locally discrete basis unless it is itself discrete. The same property holds for a
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is metrizable iff it is regular and has a basis that is countably locally discrete.
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intersecting at most one element of the collection. A collection of subsets of
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is said to be locally discrete, if each point of the space has a
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James
Munkres (1999). Topology, 2nd edition, Prentice Hall.
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4. The following is known as Bing's metrization theorem:
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68:1. Locally discrete collections are always
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