Cocountable

Definition

• $X$
  • set

\[\mathcal{T} := \{ A \subseteq X \mid A = \emptyset, \ A^{c} \text{ is countable} \} .\]

$(X, \mathcal{T})$ is called cocountable topology/countable complement topology on $X$.

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Remark

$(X, \mathcal{T})$ is top sp.

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Proposition

• $X$
  • set
• $(X, \mathcal{T})$
  • cocountable topology

(1) If $X$ is uncoutanble, the cocountable topology is not hausdorff.

proof

$\Box$

Reference

• https://en.wikipedia.org/wiki/Cocountable_topology