Proof That the Set of Real Numbers With the Cofinite Topology is Not a First Countable Space


The set of real numbers with the cofinite topology is not a first countable space.


The cofinite topologyon a setcontainsand the complements of finite sets.

Supposeis a first countable space. Letbe a countable open base atEachis open sois closed hence finite.

The setis a countable union of finite sets henceis countable andis not countable. A pointexists with

We have

Hencfor all(1)

The setis open inas a complement of an open set andsince

is a local base atHenceexists such that

Hencecontradicting (1) and setr with the cofinte topology is not first countable.