Theorems · Theorem · general topology
exists_isOpen_xor_mem
∀ {X : Type u_1} [inst : TopologicalSpace X] [T0Space X] {x y : X}, x ≠ y → ∃ U, IsOpen U ∧ Xor (x ∈ U) (y ∈ U)- Defined in
- Mathlib.Topology.Separation.Basic
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpaceT0Space
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- IsOpenstatement · cited by 2,400
- T0Spacestatement and proof · cited by 179
- Xorstatement · cited by 56
- t0Space_iff_exists_isOpen_xor_memproof · cited by 2
Cited by4
Results whose statement or proof uses this declaration.
- totallySeparatedSpace_of_t0_of_basis_clopenproof · cited by 2
- minimal_nonempty_closed_subsingletonproof · cited by 1
- minimal_nonempty_open_subsingletonproof · cited by 1
- exists_isOpen_xor'_memproof · cited by 0