first countable

Let $X$ be a topological space and let $x\in X$. $X$ is said to be 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 if for every $x\in X$, $X$ is first countable at $x$.

Remark. Equivalently, one can take each $B_n$ in the sequence to be open neighborhood of $x$.

Title	first countable
Canonical name	FirstCountable
Date of creation	2013-03-22 12:23:33
Last modified on	2013-03-22 12:23:33
Owner	Evandar (27)
Last modified by	Evandar (27)
Numerical id	5
Author	Evandar (27)
Entry type	Definition
Classification	msc 54D99
Synonym	first axiom of countability
Related topic	SecondCountable
Related topic	TestingForContinuityViaNets