Prime Cofinite Convergence (素補有限収束) (Example III.4.3)

• Prime Cofinite Convergence (素補有限収束) (Example III.4.3) #Card
  • For an infinite set $X$ and $x_\infty \in X$: $\pi = \pi[x_\infty, (X)0]$ declares each $x \neq x\infty$ isolated, and $x_\infty \in \lim_\pi \mathcal{F}$ whenever $\mathcal{F} \supset (X, x_\infty)_0$.

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Let $X$ be infinite and $x_\infty \in X$. The prime cofinite convergence $\pi = \pi[x_\infty, (X)0]$ on $X$: each $x \neq x\infty$ is isolated, and $$x_\infty \in \lim_\pi \mathcal{F} \iff \mathcal{F} \supset (X, x_\infty)0,$$ where $(X, x\infty)0$ is the cofinite filter centered at $x\infty$（cards/topology/cofinite-filter-of-b-centered-at-a）.

• Each free filter on $X$ converges to $x_\infty$（$\mathcal{F} \supset (X)0 \supset (X, x\infty)_0$）。
• A Hausdorff pretopology。$X$ 可算なら列の言葉で ${x_\infty} = \lim_\pi (x_n)_n$（一点コンパクト化 $\omega + 1$ 型の収束）。
• $X$ 非可算なら可算指標でない（Proposition II.2.19）。