Theorems · Theorem · general topology

continuous_discrete_rng

∀ {β : Type u_2} {α : Type u_3} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] [DiscreteTopology β]
  {f : α → β}, Continuous f ↔ ∀ (b : β), IsOpen (f ⁻¹' {b})

A function to a discrete topological space is continuous if and only if the preimage of every singleton is open.

Defined in

Mathlib.Topology.Order

Cited by

3 results in Mathlib

Foundations

Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpaceDiscreteTopology