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.
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.
• 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.