Prime Cocountable Convergence (素余可算収束) (Example III.4.5)

• Prime Cocountable Convergence (素余可算収束) (Example III.4.5) #Card
  • For an uncountable set $X$ and $x_\infty \in X$: the pretopology $\pi[x_\infty, (X)1]$ whose vicinity filter at the (unique) non-isolated point $x\infty$ is the cocountable filter centered at $x_\infty$.

Back

Let $X$ be an uncountable set and $x_\infty \in X$. The prime cocountable convergence $\pi[x_\infty, (X)1]$ is a pretopology, in which the vicinity filter of the non-isolated point $x\infty$ is the cocountable filter of $X$ centered at $x_\infty$: $$V_\pi(x_\infty) = (X, x_\infty)1 := {F \subset X : x\infty \in F \wedge \operatorname{card}(X \setminus F) \leq \aleph_0}.$$

参照: cards/topology/cocountable-filter-at-b。より一般に $\operatorname{card} X \geq \kappa \geq \aleph_0$ なら co-$\kappa$ フィルター（$\operatorname{card}(X \setminus F) < \kappa$）は proper。