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locally finite collection (Definition)

Let $ \mathcal{C}$ be a collection of subsets of a topological space $ X$.

$ \mathcal{C}$ is said to be locally finite if for all $ x\in X$ there is a neighbourhood $ U$ of $ x$ such that $ U \cap A = \varnothing$ for all but finitely many $ A \in \mathcal{C}$.

Similarly, $ \mathcal{C}$ is said to be locally countable if for all $ x\in X$ there is a neighbourhood $ U$ of $ x$ such that $ U \cap A = \varnothing$ for all but countably many $ A \in \mathcal{C}$.



"locally finite collection" is owned by yark. [ full author list (2) | owner history (1) ]
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See Also: point finite

Also defines:  locally finite, locally countable collection, locally countable
Keywords:  topology

Attachments:
the union of a locally finite collection of closed sets is closed (Theorem) by yark
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Cross-references: neighbourhood, topological space, subsets, collection
There are 10 references to this entry.

This is version 7 of locally finite collection, born on 2002-01-22, modified 2008-10-03.
Object id is 1542, canonical name is LocallyFinite.
Accessed 4078 times total.

Classification:
AMS MSC: 54D20 (General topology :: Fairly general properties :: Noncompact covering properties )
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