Countability axioms

First countability axiom

A topological space (𝑋,T) is first-countable iff every 𝑥 ∈𝑋 has a countable neighbourhood basis. topology

Note that since every neighbourhood contains an open neighbourhood, we can only consider open neighbourhood bases without loss of generality.

Properties of first-countable spaces

• Every point has a Nested neighbourhood basis.
• Sequential continuity and Continuity are equivalent.
• Sequential closedness and closedness are equivalent.
• Uniqueness of the limit iff hausdorff.

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