Second Countable Space

Let \left( {X,\tau } \right) be a topological space, then X is said to be second countable space, if \tau has a countable bases.

In other words, a topological space \left( {X,\tau } \right) 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 Xis finite, then member of each \tau on Xis finite. So its base is finite. Hence \left( {X,\tau } \right) is second countable space. Now we show that \left( {X,\tau } \right) is first countable. Let S is 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 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 X 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 \mathbb{R}. The real line is second countable space.
• Any uncountable set X with countable topology is not first countable and so is not second countable.

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