Product Convergence (積収束) (Definition IV.6.1)
- Product Convergence (積収束) (Definition IV.6.1) #Card
- The product convergence $\prod T$ is the coarsest convergence on $|\prod T|$ for which every projection $p_\tau$ is continuous: $\prod T := \bigvee_{\tau \in T} p_\tau^- \tau$.
Let $T$ be a set of convergences. The product convergence $\prod T = \prod_{\tau \in T} \tau$ is the coarsest convergence $\xi$ on $|\prod T|$, for which $p_\tau \in C(\xi, \tau)$ for each $\tau \in T$——射影族 ${(p_\tau, \tau)}$ に関する始収束: $$\prod T := \bigvee_{\tau \in T} p_\tau^- \tau.$$
- 収束判定（Proposition IV.6.3）: $\varphi \in \lim_{\prod T} \mathcal{F} \iff p_\tau(\varphi) \in \lim_\tau p_\tau[\mathcal{F}]$ for every $\tau \in T$——各射影が収束すれば収束。
- 2 つの場合（Example IV.4.6）: $(x,y) \in \lim_{\xi \times \upsilon} \mathcal{F} \iff p_X[\mathcal{F}] \to x$ かつ $p_Y[\mathcal{F}] \to y$。
- 積への写像の連続性（Proposition IV.6.6）: $f \in C(\xi, \prod T) \iff p_\tau \circ f \in C(\xi, \tau)$ for each $\tau$。
- 有限積の離散収束は離散（Proposition IV.6.7）だが、無限積は違う（Cantor cube）。
位相の場合 Pi.topologicalSpace（= ⨅ j, induced (fun φ => φ j) (τ j)）；nhds_pi, tendsto_pi_nhds