First-countable space

An example of a space which is not first-countable is the cofinite topology on an uncountable set (such as the real line).

First-countability is strictly weaker than second-countability. Every second-countable space is first-countable, but any uncountable discrete space is first-countable but not second-countable.

Every subspace of a first-countable space is first-countable. Any countable product of a first-countable space is first-countable, although uncountable products need not be.