Homeomorphism (同相写像) (Definition IV.2.8)
- Homeomorphism (同相写像) (Definition IV.2.8) #Card
- Convergences $\xi$ and $\tau$ are homeomorphic ($\xi \approx \tau$) if there exists a bijective $f \in C(\xi, \tau)$ with $f^{-1} \in C(\tau, \xi)$; such $f$ is a homeomorphism.
Convergences $\xi$ and $\tau$ are said to be homeomorphic, in symbols $\xi \approx \tau$, if there exists a bijective map $f \in C(\xi, \tau)$ such that $f^{-1} \in C(\tau, \xi)$. Then $f$ is called a homeomorphism; $H(\xi, \tau)$ = the set of homeomorphisms.
- 同相な収束は「同じ収束のコピー」——収束の同定は常に “up to a homeomorphism”。
- (IV.3.5) による言い換え: $\xi \approx \tau \iff \exists f$ bijective with $\xi = f^-\tau \iff f\xi = \tau$。
- 例: $\operatorname{card} X = \operatorname{card} Y$ なら $\iota_X \approx \iota_Y$, $o_X \approx o_Y$；$$0 \approx $1$（$h(x) = 1-x$）；$\varsigma+ \approx \varsigma-$（$h(x) = -x$）；$\nu_\mathbb{R} \approx \nu_\mathbb{R}|_{]a,b[}$（$\tan$ で）。
位相の場合 Homeomorph X Y（X ≃ₜ Y）