Characteristic Convergence $\chi_\mathbb{F}$ (特性収束) (Example III.7.12)

• Characteristic Convergence $\chi_\mathbb{F}$ (特性収束) (Example III.7.12) #Card
  • For a set of filters $\mathbb{F}$ on $X$ containing all principal ultrafilters and closed under refinement: $\lim_{\chi_\mathbb{F}} \mathcal{F} := X$ if $\mathcal{F} \in \mathbb{F}$, and $\emptyset$ otherwise.

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Let $\mathbb{F}$ be a set of filters on $X$ such that (1) $x^\uparrow \in \mathbb{F}$ for each $x \in X$, and (2) $\mathcal{F} \in \mathbb{F}$ and $\mathcal{F} \leq \mathcal{G}$ imply $\mathcal{G} \in \mathbb{F}$. The relation $$\lim_{\chi_\mathbb{F}} \mathcal{F} := \begin{cases} X & \mathcal{F} \in \mathbb{F}, \ \emptyset & \mathcal{F} \notin \mathbb{F}, \end{cases}$$ is a convergence, called a characteristic convergence of $\mathbb{F}$.

• $\mathbb{F}$ が有限 meet で閉なら $\chi_\mathbb{F}$ は有限安定；任意の空でない部分族の meet で閉なら前位相。
• $\mathbb{F}$ はいわゆる Cauchy structure（E. Lowen-Colebunders）。