Skip to content
Journals Garden

Finitely Stable Convergence (有限安定収束) (Definition III.1.5)

  • Finitely Stable Convergence (有限安定収束) (Definition III.1.5) #Card
    • A convergence $\xi$ is finitely stable if for each $\mathcal{F}_0, \mathcal{F}1 \in \mathbb{F}|\xi|$: $\lim\xi \mathcal{F}0 \cap \lim\xi \mathcal{F}1 \subset \lim\xi (\mathcal{F}_0 \cap \mathcal{F}_1)$.

Back

Section titled “Back”

A convergence $\xi$ is called finitely stable if for each $\mathcal{F}_0, \mathcal{F}1 \in \mathbb{F}|\xi|$, $$\lim\xi \mathcal{F}0 \cap \lim\xi \mathcal{F}1 \subset \lim\xi (\mathcal{F}_0 \cap \mathcal{F}_1).$$

  • The converse inclusion always holds by (isotone); the property extends by induction to arbitrary finite collections.
  • Pointwise form: $\mathcal{F}_0, \mathcal{F}_1 \in \xi^-(x) \implies \mathcal{F}_0 \wedge \mathcal{F}_1 \in \xi^-(x)$.
  • $\iota$, $o$, $\sigma_\mathbb{R}$, $\nu_\mathbb{R}$ are all finitely stable; most natural convergences are. Some authors include finite stability in the definition of convergence.