instOrderClosedTopology — Mathlib

English

General set-theoretic lemmas about when a point lies in the image of a function, or in an interval, or in a Cartesian image, express basic logical equivalences. For instance, membership in the image of a set is equivalent to the existence of a preimage in the domain, and membership in a closed interval Icc(a,b) is equivalent to the left and right inequalities.

Русский

Обобщённые леммы теории множеств: членство в образе функции эквивалентно существованию прообраза, а членство в замкнутом промежутке Icc(a,b) эквивалентно неравенствам слева и справа.

LaTeX

$$$\\forall f:\\alpha\\to\\beta, s\\subseteq\\alpha, y\\in\\beta:\\; y\\in\\mathrm{image}(f,s) \\iff \\exists x\\in s:\\; f(x)=y.$\n\n\\forall a,b:\\; x\\in Icc(a,b) \\iff a\\le x \\le b.$$$

Lean4

instance : OrderClosedTopology A where isClosed_le' := isClosed_le_of_isClosed_nonneg isClosed_nonneg

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