Theorems · Definition · general topology
IsClopen
{X : Type u} → [TopologicalSpace X] → Set X → PropA set is clopen if it is both closed and open.
- Defined in
- Mathlib.Topology.Defs.Basic
- Cited by
- 189 results in Mathlib
- Foundations
- Depth 3 from the axioms, rests on 7 definitions · uses no axioms
- Assumes
- TopologicalSpace
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.
- Setstatement and proof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- IsOpenproof · cited by 2,400
- IsClosedproof · cited by 1,639
Cited by205
Results whose statement or proof uses this declaration.
- IsClopen.isOpenstatement and proof · cited by 21
- IsClopen.complstatement and proof · cited by 12
- IsClopen.isClosedstatement and proof · cited by 12
- IsClopen.preimagestatement and proof · cited by 10
- isClopen_discretestatement · cited by 8
- LocallyConstant.ofIsClopenstatement and proof · cited by 7
- isClopen_iffstatement and proof · cited by 6
- isClopen_iff_frontier_eq_emptystatement · cited by 6
- LocallyConstant.indicatorstatement and proof · cited by 6
- TopologicalSpace.Clopens.isClopen'statement · cited by 5
- ConnectedComponents.equivOfIsClopenstatement and proof · cited by 5
- isTopologicalBasis_isClopenstatement and proof · cited by 5
Showing the 200 most cited of 205.