Royal Road to Topology Chapter IV
Chapter 4: Continuity
Section titled “Chapter 4: Continuity”「連続 = 収束を保つ」の一行から、始収束・終収束・積・商・埋め込みまでを随伴公式 (IV.3.5) 一本で組み立てる章。必須ロードマップ（II–IV）の最終章。
IV.1. Images and Preimages
Section titled “IV.1. Images and Preimages”- cards/topology/order-on-families
- cards/topology/image-preimage-adjunction
- cards/topology/prop-iv-1-3
- cards/topology/prop-iv-1-4
IV.2. Continuous Maps
Section titled “IV.2. Continuous Maps”- cards/topology/continuous-map
- cards/topology/prop-iv-2-2
- cards/topology/continuity-epsilon-delta
- cards/topology/continuity-of-metric-and-norm
- cards/topology/lem-iv-2-5
- cards/topology/prop-iv-2-6
- cards/topology/homeomorphism
IV.3. Continuity and Order
Section titled “IV.3. Continuity and Order”- cards/topology/initial-convergence
- cards/topology/final-convergence
- cards/topology/quotient-map
- cards/topology/subconvergence
- cards/topology/embedding
- cards/topology/prop-iv-3-5
- cards/topology/prop-iv-3-14
- cards/topology/prop-iv-3-15
IV.4. Initial Convergences
Section titled “IV.4. Initial Convergences”- cards/topology/initial-convergence-family
- cards/topology/lem-iv-4-3
- cards/topology/prop-iv-4-5
- cards/topology/sigma-initial
IV.5. Selections and Products
Section titled “IV.5. Selections and Products”IV.6. Product Convergences
Section titled “IV.6. Product Convergences”- cards/topology/product-convergence
- cards/topology/polyhedral-filter
- cards/topology/cantor-cube
- cards/topology/pointwise-convergence
- cards/topology/prop-iv-6-11
- cards/topology/prop-iv-6-12
IV.7. Diagonal Product Maps
Section titled “IV.7. Diagonal Product Maps”IV.8. Final Convergences
Section titled “IV.8. Final Convergences”IV.9. *Lower Semicontinuity
Section titled “IV.9. *Lower Semicontinuity”IV.10. Supplement
Section titled “IV.10. Supplement”[!IMPORTANT] 随伴公式 (IV.3.5) がこの章のすべて $$f\xi \geq \tau \iff f \in C(\xi, \tau) \iff \xi \geq f^-\tau$$ 連続性の議論を不等式の計算に変える。ここから:
構成 定義 普遍性 始収束（族） $\bigvee_j f_j^-\tau_j$ $h \in C(\theta,\xi) \iff f_j \circ h \in C(\theta,\tau_j)$ 積収束 $\bigvee_\tau p_\tau^-\tau$（射影の始収束） $f \in C(\xi,\prod T) \iff p_\tau \circ f \in C(\xi,\tau)$ 部分収束 $j_V^-\xi \approx \xi\vert_V$ — 終収束（族） $\bigwedge_j f_j\xi_j$ $h \in C(\tau,\upsilon) \iff h \circ f_j \in C(\xi_j,\upsilon)$ 余積 $\bigoplus_j \tau_j = \bigwedge_j i_j\tau_j$ — 商 $f\xi$（$f$ 全射） —
[!NOTE] 本書 ↔ Mathlib 規約ブリッジ（連続性編）
本書 Mathlib（位相の場合） $f \in C(\xi, \tau)$ Continuous f$f[\mathcal{F}]$ / $f^-[\mathcal{G}]$ Filter.map f F/Filter.comap f G$f^-\tau$（始収束） TopologicalSpace.induced f τ$f\xi$（終収束） TopologicalSpace.coinduced f ξ(IV.3.5) gc_coinduced_induced（Galois 接続）(IV.1.4) $f^-[\mathcal{H}] \leq \mathcal{G} \iff \mathcal{H} \leq f[\mathcal{G}]$ Filter.map_le_iff_le_comap（map ⊣ comap）積収束 / 各点収束 Pi.topologicalSpace/tendsto_pi_nhds
- 反例の要点: 連続性は一般に列では判定できない（$\nu_\mathbb{R}$ と $\sigma_\mathbb{R}$ は列上一致するのに $i_\mathbb{R} \notin C(\nu_\mathbb{R}, \sigma_\mathbb{R})$、Example IV.2.7）。距離化可能なら列で十分（Prop IV.2.6）。
- 商・終収束は前位相性を保たない（区間の端点貼り合わせ Example IV.3.10、fan Example IV.3.13、Example IV.10.15）→ 第VI章 pretopologies での修正子の伏線。
- 測度論への視線: polyhedral filter（筒集合）→ 積測度・Kolmogorov 拡張の舞台；各点収束 $\sigma^J$ → 分布収束・特性関数；選択原理（IV.9.4）→ 可測選択定理の位相版原型。
- 演習: IV.10.4（解答付き）、IV.10.8（解答付き：$f(\bigwedge \xi) = \bigwedge f\xi$ 等の 4 公式）、IV.10.9（$\rho \not\approx \mu$ を示す）は書籍未解答だが自力証明済み ✅（exercises/chapter4.md 参照）、IV.10.13（解答付き：$(\prod \xi_j)|_{\triangle(X)} = \bigvee \xi_j$）。