# Second Countable Space

Let be a topological space, then is said to be the second countable space, if has a countable base.

In other words, a topological space 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 is finite, then a member of each on is finite. So its base is finite. Hence is the second countable space. Now we show that is the first countable space. Let be a subbase of . So, (Countable), then (countable), so is also countable.

Therefore, 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 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 . The real line is the second countable space.

• Any uncountable set with a countable topology is not the first countable and so is not the second countable space.