# Second Countable Space

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

In other words, a topological space $\left( {X,\tau } \right)$ is said to be 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 member of each $\tau$on $X$is finite. So its base is finite. Hence $\left( {X,\tau } \right)$ is second countable space. Now we show that $\left( {X,\tau } \right)$ is first countable. Let $S$ is 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 second countable space, as each local base is also countable, so this is also first countable space.

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