Theorems · Definition · general topology
TopologicalSpace.induced
{X : Type u_1} → {Y : Type u_2} → (X → Y) → TopologicalSpace Y → TopologicalSpace XGiven f : X → Y and a topology on Y,
the induced topology on X is the collection of sets
that are preimages of some open set in Y.
This is the coarsest topology that makes f continuous.
- Defined in
- Mathlib.Topology.Defs.Induced
- Cited by
- 148 results in Mathlib
- Foundations
- Depth 64 from the axioms, rests on 697 definitions · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Set.preimageproof · cited by 4,946
- IsOpenproof · cited by 2,400
Cited by156
Results whose statement or proof uses this declaration.
- UniformSpace.comapproof · cited by 61
- continuous_induced_domstatement · cited by 47
- continuous_induced_rngstatement and proof · cited by 38
- nhds_inducedstatement · cited by 32
- Topology.IsInducing.eq_inducedstatement · cited by 31
- induced_composestatement · cited by 26
- Topology.IsInducing.compproof · cited by 17
- continuous_iff_le_inducedstatement · cited by 16
- isOpen_inducedstatement · cited by 13
- induced_infstatement · cited by 12
- Topology.IsInducing.of_compproof · cited by 11
- Topology.IsInducing.inducedstatement · cited by 10