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Initial Convergence w.r.t. a Family (写像族に関する始収束) (Definition IV.4.1)

  • Initial Convergence w.r.t. a Family (写像族に関する始収束) (Definition IV.4.1) #Card
    • The coarsest convergence $\xi$ on $X$ for which $f_j \in C(\xi, \tau_j)$ for each $j \in J$; it exists and equals $\xi = \bigvee_{j \in J} f_j^- \tau_j$.

Back

Section titled “Back”

Let $X$ be a set, $T = {\tau_j : j \in J}$ convergences, and $f_j : X \to |\tau_j|$. The coarsest convergence $\xi$ on $X$, for which $f_j \in C(\xi, \tau_j)$ for each $j \in J$, is called the initial convergence with respect to ${(f_j, \tau_j) : j \in J}$. It exists (Proposition IV.4.2): $$\xi = \bigvee_{j \in J} f_j^- \tau_j. \tag{IV.4.2}$$

  • 各写像に関する始収束たちの sup。積収束は射影族に関する始収束（cards/topology/product-convergence）。
  • 収束の sup 自体が恒等写像族に関する始収束（Remark IV.4.4）。

Lean

Section titled “Lean”

位相の場合 ⨅ j, induced (f j) (τ j)（Mathlib は順序が逆なので inf）