Skip to content
Journals Garden

Chaotic Convergence (混沌収束 / 密着収束) (Example III.1.2)

  • Chaotic Convergence (混沌収束 / 密着収束) (Example III.1.2) #Card
    • The convergence $o = o_X$ (also called antidiscrete) defined by $X = \lim_o \mathcal{F}$ for every $\mathcal{F} \in \mathbb{F}X$: each (proper) filter converges to each point.

Back

Section titled “Back”

The convergence $o = o_X$ on $X$ (also called antidiscrete) defined by $X = \lim_o \mathcal{F}$ for every $\mathcal{F} \in \mathbb{F}X$, that is, each (proper) filter $o$-converges to each point.

  • $o_X$ is the coarsest convergence on $X$ (Proposition III.3.2) and a pretopology.
  • $o_X$ is free (alias $T_1$) $\iff \operatorname{card} X = 1$; for $\operatorname{card} X > 1$ it is not even $T_0$.

Meaning

Section titled “Meaning”

離散収束 $\iota$ と対をなす最も単純な収束。各 proper filter が各点に $o$-収束する——antidiscrete convergence とも呼ばれる。

$o$ は finitely stable かつ pretopology である。card $X > 1$ のとき $o_X$ は $T_0$ でさえない（card $X = 1$ なら $\iota_X = o_X$ となり任意の収束と一致する）。

出典: refs/math/topology/royal-road-to-topology/pdfs/chp3-2024-convergence-of-filters.pdf p.36

Lean

Section titled “Lean”

位相の場合は ⊤ : TopologicalSpace X（密着位相、𝓝 x = ⊤）。Notes/ChapterIII.lean : chaotic