Theorems · Theorem · general topology
continuous_iff_preimage_interior_subset_interior_preimage
∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},
Continuous f ↔ ∀ (s : Set Y), f ⁻¹' interior s ⊆ interior (f ⁻¹' s)- Defined in
- Mathlib.Topology.Continuous
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 65 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites11
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.preimagestatement and proof · cited by 4,946
- Continuousstatement and proof · cited by 2,592
- IsOpenproof · cited by 2,400
- le_reflproof · cited by 2,061
- interiorstatement and proof · cited by 714
- le_imp_le_of_le_of_leproof · cited by 576
- IsOpen.interior_eqproof · cited by 58
- preimage_interior_subset_interior_preimageproof · cited by 14
- subset_interior_iff_isOpenproof · cited by 6
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