# First Countable Space

Let $\left( {X,\tau } \right)$ be a topological space, then $X$ is said to be the first countable space if for every $x \in X$ has a countable local base; i.e., if every $x \in X$, ${{\rm B}_x}$ is countable.

In other words, a topological space $\left( {X,\tau } \right)$ is said to be the first countable space if every point $x$ of $X$ has a countable neighbohood base. A first countable space is also said to be a space satisfying the first axiom of countability.

Example:

If $X$ is finite, then $\left( {X,\tau } \right)$ is first countable space. As $X$ is finite, all of its subsets are finite. If ${{\rm B}_x}$ is a local base of $x \in X$, then ${{\rm B}_x}$ is also finite. So, $\left( {X,\tau } \right)$ is the first countable space.

Example:
Let $X = \left\{ {a,b,c,d} \right\}$ be a non-empty set and $\tau = \left\{ {\phi ,X,\left\{ a \right\},\left\{ b \right\},\left\{ {a,b} \right\},\left\{ {c,d} \right\},\left\{ {a,c,d} \right\},\left\{ {b,c,d} \right\}} \right\}$ be a topology defined on $X$.

Base at $a$ $= {{\rm B}_a} = \left\{ a \right\}$
Base at $b$ $= {{\rm B}_b} = \left\{ b \right\}$
Base at $c$ $= {{\rm B}_c} = \left\{ {c,d} \right\}$
Base at $d$ $= {{\rm B}_d} = \left\{ {c,d} \right\}$

Here each local base is countable, so $\left( {X,\tau } \right)$ is the first countable space.

Example:

If $X$ is either countable or uncountable and $P\left( X \right)$ is a discrete topology on $X$, then $\left( {X,\tau } \right)$ is always the first countable space, because for each $x \in X$, ${{\rm B}_x}$ (singleton) is the local base and so the local base is finite. ${{\rm B}_x} = \left\{ {\left\{ x \right\}} \right\}$, $x \in {{\rm B}_x} \subseteq U$ and ${{\rm B}_x}$ is countable (finite).