English
Suppose F preserves and reflects preregular structure and the prerequisites hold (C preregular, D preregular, and F preserves and reflects preregular aspects). Then the regular topology on C coincides with the induced topology of the regular topology on D along F: regularTopology C = F.inducedTopology (regularTopology D).
Русский
Пусть F: C → D сохраняет и отражает предподобие прeregular, а также выполняются необходимые условия. Тогда регулярная топология на C совпадает с индуцированной топологией регулярной топологии D вдоль F: regularTopology(C) = F.inducedTopology(regularTopology(D)).
LaTeX
$$$\operatorname{regularTopology}(C) = F\.inducedTopology(\operatorname{regularTopology}(D))$$$
Lean4
theorem eq_induced :
haveI := F.reflects_preregular
regularTopology C = F.inducedTopology (regularTopology _) :=
by
ext X S
have := F.reflects_preregular
rw [← exists_effectiveEpi_iff_mem_induced F X]
rw [← mem_sieves_iff_hasEffectiveEpi S]
