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).