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Coproduct / Sum of Convergences (余積・和収束) (Definition IV.8.2)

  • Coproduct / Sum of Convergences (余積・和収束) (Definition IV.8.2) #Card
    • For pairwise disjoint ${\tau_j}$, the coproduct $\bigoplus_{j \in J} \tau_j := \bigwedge_{j \in J} i_j \tau_j$ — the final convergence with respect to the injections $i_j : |\tau_j| \to \coprod_k |\tau_k|$.

Back

Section titled “Back”

Let ${\tau_j : j \in J}$ be convergences with pairwise disjoint underlying sets. The coproduct (also called the sum) is the final convergence with respect to the family of injections $i_j : |\tau_j| \to \coprod_{k \in J} |\tau_k|$: $$\bigoplus_{j \in J} \tau_j := \bigwedge_{j \in J} i_j \tau_j.$$

  • 余積からの写像の連続性 = 各 injection との合成の連続性。
  • 例（Example IV.3.13）: prime 収束の和 $\bigoplus_n \xi_n$（前位相）から極を一点に貼り合わせる写像の終収束として sequential fan が得られる——「終収束は前位相性を壊しうる」。

Lean

Section titled “Lean”

位相の場合 Sigma.topologicalSpace（instTopologicalSpaceSum は 2 項版）