Prime Convergence (素収束) (Definition III.4.1)
- Prime Convergence (素収束) (Definition III.4.1) #Card
- A convergence is prime if it has at most one non-isolated point, called the pole.
A convergence is called prime if it has at most one non-isolated point, which is called the pole of that convergence. If $x_\infty$ is a pole of a prime convergence $\xi$, we say $\xi$ is prime at $x_\infty$.
- 記法: $\pi[x, \mathbb{D}]$ は ${\mathcal{D} \wedge x^\uparrow : \mathcal{D} \in \mathbb{D}}$ を極 $x$ での pavement とする prime convergence。$\mathbb{D} = {\mathcal{D}}$ のときは $\pi[x, \mathcal{D}]$ と略記。
- A prime convergence $\xi$ is a pretopology $\iff \xi = \pi[x, V_\xi(x)]$ for some $x$.
- $\iota_X$ は prime；$o_X$ は $X$ が一点のときだけ prime。
- prime かつ free ($T_1$) $\implies$ Hausdorff（cards/topology/prop-iii-4-2）。