/-
Copyright 2025 The Formal Conjectures Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
https://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/
module
public import Mathlib.Topology.Homeomorph.Lemmas@[expose] public section/- A bijection between discrete topological spaces induces a homeomorphism. -/
def Homeomorph.OfDiscrete {X Y : Type*} [TopologicalSpace X] [DiscreteTopology X]
[TopologicalSpace Y] [DiscreteTopology Y] (f : X ≃ Y) : X ≃ₜ Y where
toEquiv := f
continuous_toFun := continuous_of_discreteTopology
continuous_invFun := continuous_of_discreteTopologyA bijection between discrete topological spaces is a homeomorphism.
theorem IsHomeomorph.equiv_of_discreteTopology {X Y : Type*} [TopologicalSpace X]
[DiscreteTopology X] [TopologicalSpace Y] [DiscreteTopology Y] (f : X ≃ Y) : IsHomeomorph f :=
(Homeomorph.OfDiscrete f).isHomeomorph