Theorems · Theorem · general topology
exists_infinite_discreteTopology
∀ (X : Type u_1) [inst : TopologicalSpace X] [T2Space X] [Infinite X], ∃ s, s.Infinite ∧ DiscreteTopology ↑s
If X is an infinite Hausdorff topological space, then there exists an infinite set s : Set X
that has the induced topology is the discrete topology.
- Defined in
- Mathlib.Topology.NatEmbedding
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites16
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
- Set.Elemstatement · cited by 7,166
- Set.rangeproof · cited by 4,705
- T2Spacestatement and proof · cited by 1,351
- DiscreteTopologystatement · cited by 373
- Homeomorph.symmproof · cited by 365
- Infinitestatement and proof · cited by 352
- Topology.IsEmbeddingproof · cited by 294
- Set.Infinitestatement · cited by 263
- Topology.IsEmbedding.injectiveproof · cited by 103
- Homeomorph.isEmbeddingproof · cited by 73
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