Theorems · Theorem · functional analysis

WithSeminorms.firstCountableTopology

∀ {𝕜 : Type u_2} {E : Type u_6} {ι : Type u_9} [inst : NontriviallyNormedField 𝕜] [inst_1 : AddCommGroup E]
  [inst_2 : Module 𝕜 E] [Countable ι] {p : SeminormFamily 𝕜 E ι} [inst_4 : TopologicalSpace E],
  WithSeminorms p → FirstCountableTopology E

If the topology of a space is induced by a countable family of seminorms, then the topology is first countable.

Defined in

Mathlib.Analysis.LocallyConvex.WithSeminorms

Cited by

0 results in Mathlib

Foundations

Depth 161 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldAddCommGroupModuleCountableTopologicalSpace