# 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 FirstCountable 2013-03-22 12:23:33 2013-03-22 12:23:33 Evandar (27) Evandar (27) 5 Evandar (27) Definition msc 54D99 first axiom of countability SecondCountable TestingForContinuityViaNets