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first countable (Definition)

Let $ X$ be a topological space and let $ x\in X$. $ X$ is said to be first countable at $ x$ if there is a sequence $ (B_n)_{n\in\mathbb{N}}$ of open sets such that whenever $ U$ is an open set containing $ x$, there is $ n\in\mathbb{N}$ such that $ x\in B_n\subseteq U$.

The space $ X$ is said to be first countable if for every $ x\in X$, $ X$ is first countable at $ x$.



"first countable" is owned by Evandar.
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See Also: second countable

Other names:  first axiom of countability
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Cross-references: open sets, sequence, topological space
There are 9 references to this entry.

This is version 1 of first countable, born on 2002-02-19.
Object id is 2187, canonical name is FirstCountable.
Accessed 4239 times total.

Classification:
AMS MSC: 54D99 (General topology :: Fairly general properties :: Miscellaneous)
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