First-countable space

In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space ${\displaystyle X}$ is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point ${\displaystyle x}$ in ${\displaystyle X}$ there exists a sequence ${\displaystyle N_{1},N_{2},\ldots }$ of neighbourhoods of ${\displaystyle x}$ such that for any neighbourhood ${\displaystyle N}$ of ${\displaystyle x}$ there exists an integer ${\displaystyle i}$ with ${\displaystyle N_{i}}$ contained in ${\displaystyle N.}$ Since every neighborhood of any point contains an open neighborhood of that point, the neighbourhood basis can be chosen without loss of generality to consist of open neighborhoods.