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.