Let be a topological space, then is said to be first countable space if for every  has a countable local bases, i.e., if every ,  is countable.

            In other words, a topological space  is said to be the first countable space if every point  of has a countable neighbourhood base. A first countable space is also said to be a space satisfying the first axiom of countability.

Example:
            If is finite, then is first countable space. As is finite, so its every subset is finite. If  is local base of , then  is also finite. So,  is first countable space.

Example:
            Let  be a non-empty set and  be topology defined on .
            Base at “a”
            Base at “b”
            Base at “c”
            Base at “d”
Here each local base is countable, so  is first countable space.

Example:
            If is either countable or uncountable and is discrete topology on , then  always first countable space, because for each ,  (singleton) is the local base and so local base is finite. ,  and  is countable (finite).