Theorems · Theorem · general topology

IsProperMap.universally_closed

∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y} (Z : Type u_6)
  [inst_2 : TopologicalSpace Z], IsProperMap f → IsClosedMap (Prod.map f id)

A proper map f : X → Y is universally closed: for any topological space Z, the map Prod.map f id : X × Z → Y × Z is closed. We will prove in isProperMap_iff_universally_closed that proper maps are exactly continuous maps which have this property, but this result should be easier to use because it allows Z to live in any universe.

Defined in

Mathlib.Topology.Maps.Proper.Basic

Cited by

3 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpaceTopologicalSpace