Theorems · Theorem · general topology

Topology.IsInducing.isSeparable_preimage

∀ {β : Type v} {α : Type u_2} [inst : TopologicalSpace α] [TopologicalSpace.PseudoMetrizableSpace α] {f : β → α}
  [inst_2 : TopologicalSpace β],
  Topology.IsInducing f → ∀ {s : Set α}, TopologicalSpace.IsSeparable s → TopologicalSpace.IsSeparable (f ⁻¹' s)

The preimage of a separable set by an inducing map is separable.

Defined in

Mathlib.Topology.MetricSpace.Pseudo.Basic

Cited by

1 results in Mathlib

Foundations

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

Assumes

TopologicalSpaceTopologicalSpace.PseudoMetrizableSpaceTopologicalSpace