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Diagonal Map (対角写像) (Definition IV.5.1)

  • Diagonal Map (対角写像) (Definition IV.5.1) #Card
    • The diagonal map $\triangle : X \to X^J = \prod_{j \in J} X$ is defined by $\triangle(x)(j) := x$ for each $j \in J$.

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Section titled “Back”

The diagonal map $$\triangle : X \to X^J = \prod_{j \in J} X$$ is defined by $\triangle(x)(j) := x$ for each $j \in J$（$x$ を定値写像 $j \mapsto x$ に送る）.

  • The image $\triangle(X)$ is the diagonal subset: $\varphi \in \triangle(X) \iff \exists x\ \forall j,\ \varphi(j) = x$。
  • $J$ が 2 点なら $\triangle(X) = \Delta_X = {(x,x) : x \in X}$（恒等関係、(I.1.3)）。
  • Exercise IV.10.13: $(\prod_{j \in J} \xi_j)|{\triangle(X)} = \bigvee{j \in J} \xi_j$——収束の sup は積の対角線。