# Second Countable Space

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

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

Therefore, 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 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 . The real line is second countable space.

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