# First Countable Space

Let be a topological space, then is said to be the first countable space if for every has a countable local base; 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 neighbohood 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, all of its subsets are finite. If is a local base of , then is also finite. So, is the first countable space.

**Example:**

Let be a non-empty set and be a topology defined on .

Base at

Base at

Base at

Base at

Here each local base is countable, so is the first countable space.

**Example:**

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