Theorems · Theorem · general topology

isClosedMap_iff_clusterPt

∀ {X : Type u_1} {Y : Type u_2} {f : X → Y} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y],
  IsClosedMap f ↔
    ∀ (s : Set X) (y : Y), MapClusterPt y (Filter.principal s) f → ∃ x, f x = y ∧ ClusterPt x (Filter.principal s)

A map f : X → Y is closed if and only if for all sets s, any cluster point of f '' s is the image by f of some cluster point of s. If you require this for all filters instead of just principal filters, and also that f is continuous, you get the notion of proper map. See isProperMap_iff_clusterPt.

Defined in

Mathlib.Topology.Maps.Basic

Cited by

1 results in Mathlib

Foundations

Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpace