Theorems · Theorem · general topology
isCompact_iff_compactSpace
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X}, IsCompact s ↔ CompactSpace ↑s- Defined in
- Mathlib.Topology.Compactness.Compact
- Cited by
- 50 results in Mathlib
- Foundations
- Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- Set.Elemstatement · cited by 7,166
- IsCompactstatement · cited by 1,282
- CompactSpacestatement · cited by 593
- isCompact_univ_iffproof · cited by 9
- isCompact_iff_isCompact_univproof · cited by 5
Cited by50
Results whose statement or proof uses this declaration.
- IsClosed.vadd_left_of_isCompactproof · cited by 5
- tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compactproof · cited by 5
- IsClosed.smul_left_of_isCompactproof · cited by 5
- NonUnitalContinuousFunctionalCalculus.isCompact_quasispectrumproof · cited by 4
- ContinuousFunctionalCalculus.isCompact_spectrumproof · cited by 3
- Summable.hasProdUniformlyOn_one_addproof · cited by 3
- AlgebraicGeometry.exists_map_preimage_le_map_preimageproof · cited by 3
- isCompact_setOfPred_finiteMeasure_le_of_isCompactproof · cited by 2
- loc_compact_Haus_tot_disc_of_zero_dimproof · cited by 2
- ArzelaAscoli.isCompact_of_equicontinuousproof · cited by 2
- EquicontinuousOn.comap_uniformOnFun_eqproof · cited by 2
- AlgebraicGeometry.exists_app_map_eq_zero_of_isLimitproof · cited by 2