# Second Countable Space

Let $\left( {X,\tau } \right)$ be a topological space, then $X$ is said to be the second countable space, if $\tau$ has a countable base.

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

Example:

If $X$ is finite, then a member of each $\tau$on $X$ is finite. So its base is finite. Hence $\left( {X,\tau } \right)$ is the second countable space. Now we show that $\left( {X,\tau } \right)$ is the first countable space. Let $S$ be a subbase of $\tau$. So, $S \subseteq P\left( X \right)$ (Countable), then ${\rm B} \subseteq P\left( X \right)$ (countable), so ${\rm B}$ is also countable.

Therefore, $\left( {X,\tau } \right)$ is the second countable space, as each local base is also countable, so this is also the first countable space.

Theorems
• Every second countable space is the first countable space, but the converse may not be true.
• Any uncountable set $X$ with a co-finite topology is not the first countable space and so it is not the second countable space.
• The set of all intervals with rational ends is a countable base for the usual topology on $\mathbb{R}$. The real line is the second countable space.
• Any uncountable set $X$ with a countable topology is not the first countable and so is not the second countable space.