isClosedMap_fst_of_compactSpace — Mathlib

English

The preimage of a compact set under Prod.fst is compact, using that Y is compact and the swap map.

Русский

Обратное изображение компактного множества под Prod.fst компактно.

LaTeX

$$IsClosedMap(Prod.fst) := IsClosedMap Prod.fst$$

Lean4

/-- If `Y` is a compact topological space, then `Prod.fst : X × Y → X` is a closed map. -/
theorem isClosedMap_fst_of_compactSpace [CompactSpace Y] : IsClosedMap (Prod.fst : X × Y → X) :=
  isClosedMap_snd_of_compactSpace.comp isClosedMap_swap

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