Quotient Map (商写像) (Definition IV.3.9)
- Quotient Map (商写像) (Definition IV.3.9) #Card
- A surjective $f : X \to Y$ is a convergence quotient map if $\tau$ is the finest convergence with $f \in C(\xi, \tau)$ (i.e. $\tau = f\xi$); finitely stable / pretopologically quotient if $\tau$ is the finest finitely stable convergence / pretopology with $f \in C(\xi, \tau)$.
Let $\xi$ on $X$, $\tau$ on $Y$. A surjective map $f : X \to Y$ is called:
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convergence quotient map if $\tau$ is the finest convergence such that $f \in C(\xi, \tau)$（このとき $\tau = f\xi$ が quotient convergence）;
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finitely stable quotient map if $\tau$ is the finest finitely stable convergence with $f \in C(\xi, \tau)$;
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pretopologically quotient map if $\tau$ is the finest pretopology with $f \in C(\xi, \tau)$（quotient pretopology）。
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全射でも quotient とは限らない: $\xi > \tau$ で $i_X \in C(\xi, \tau)$ だが $i\xi = \xi > \tau$。
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例（Example IV.3.10）: $\mu = \nu|{[0,1]}$、$h(1) := 0$ で端点を貼り合わせると、$h\mu$（quotient convergence）は前位相でない。$V\tau(0) := V_\mu(0) \wedge (V_\mu(1) \vee [0,1[)$ とした $\tau$ は quotient pretopology で、円周 $\rho = j_S^-(\nu_\mathbb{R}^2)$ と同相（Example IV.3.16）。
位相の場合 Topology.IsQuotientMap f（coinduced が一致する全射）