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 X is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point x in X there exists a sequence U₁, U₂, … of open neighbourhoods of x such that for any open neighbourhood V of x there exists an integer i with U_i contained in V.