Theorems · Theorem · general topology

isProperMap_iff_isClosedMap_filter

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

A map f : X → Y is proper if and only if it is continuous and the map (Prod.map f id : X × Filter X → Y × Filter X) is closed. This is stronger than isProperMap_iff_universally_closed since it shows that there's only one space to check to get properness, but in most cases it doesn't matter.

Defined in

Mathlib.Topology.Maps.Proper.UniversallyClosed

Cited by

0 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpace