# 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.