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Sorgenfrey Line (ゾルゲンフライ直線) (Example III.3.6)

  • Sorgenfrey Line (ゾルゲンフライ直線) (Example III.3.6) #Card
    • The convergence $\varsigma = \varsigma_+$ on $\mathbb{R}$: $x \in \lim_\varsigma \mathcal{F}$ whenever $[x, x+\varepsilon[ \in \mathcal{F}$ for each $\varepsilon > 0$ (right half-open interval convergence).

Back

Section titled “Back”

The Sorgenfrey convergence $\varsigma = \varsigma_+$ on the real line: $x \in \lim_\varsigma \mathcal{F}$ whenever $[x, x + \varepsilon[ \in \mathcal{F}$ for each $\varepsilon > 0$. The coarsest filter converging to $x$ is generated by ${[x, x+2^{-n}[ ,: n < \omega}$.

  • A Hausdorff pretopology of countable character; $\varsigma > \nu_\mathbb{R}$.
  • Twin: 左半開区間収束 $\varsigma_-$（$]x-\varepsilon, x] \in \mathcal{F}$）。$\varsigma_+$ と $\varsigma_-$ は比較不能。
  • 有限安定収束の中での極値（Exercise III.7.3）: $$\varsigma_+ \wedge \varsigma_- = \nu_\mathbb{R}, \qquad \varsigma_+ \vee \varsigma_- = \iota.$$

Lean

Section titled “Lean”

Mathlib: Counterexamples/SorgenfreyLine.lean の SorgenfreyLine（記法 ℝₗ）