Theorems · Theorem · functional analysis

WithSeminorms.T1_of_separating

∀ {𝕜 : Type u_2} {E : Type u_6} {ι : Type u_9} [inst : NormedField 𝕜] [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E]
  [inst_3 : TopologicalSpace E] {p : SeminormFamily 𝕜 E ι},
  WithSeminorms p → (∀ (x : E), x ≠ 0 → ∃ i, (p i) x ≠ 0) → T1Space E

A separating family of seminorms induces a T₁ topology.

Defined in

Mathlib.Analysis.LocallyConvex.WithSeminorms

Cited by

1 results in Mathlib

Foundations

Depth 166 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedFieldAddCommGroupModuleTopologicalSpace