Theorems · Theorem · general topology

isProperMap_iff_universally_closed

∀ {X : Type u} {Y : Type v} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},
  IsProperMap f ↔ Continuous f ∧ ∀ (Z : Type u) [inst_2 : TopologicalSpace Z], IsClosedMap (Prod.map f id)

A map f : X → Y is proper if and only if it is continuous and universally closed, in the sense that for any topological space Z, the map Prod.map f id : X × Z → Y × Z is closed. Note that Z lives in the same universe as X here, but IsProperMap.universally_closed does not have this restriction. This is taken as the definition of properness in [N. Bourbaki, General Topology][bourbaki1966].

Defined in

Mathlib.Topology.Maps.Proper.UniversallyClosed

Cited by

0 results in Mathlib

Foundations

Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpace