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Theorems · Theorem · general topology

IsConnected.subset_connectedComponent

∀ {α : Type u} [inst : TopologicalSpace α] {x : α} {s : Set α}, IsConnected s → x ∈ s → s ⊆ connectedComponent x
Defined in
Mathlib.Topology.Connected.Basic
Cited by
6 results in Mathlib
Foundations
Depth 9 from the axioms · uses no axioms
Assumes
TopologicalSpace
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

connectedComponent_eq · cited by 6connectedComponent_eqContinuous.image_connectedComponent_subset · cited by 6Continuous.image_connecte…isClosed_connectedComponent · cited by 3isClosed_connectedCompone…Topology.IsCoinducing.preimage_connectedComponent · cited by 2IsCoinducing.preimage_con…pathComponent_subset_component · cited by 2pathComponent_subset_comp…irreducibleComponent_subset_connectedComponent · cited by 0irreducibleComponent_subs…Set · cited by 53352SetTopologicalSpace · cited by 24529TopologicalSpaceIsConnected · cited by 116IsConnectedconnectedComponent · cited by 68connectedComponentIsPreconnected.subset_connectedComponent · cited by 10IsPreconnected.subset_con…IsConnected.subset_connectedC…CITED BYCITES

Cites5

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Cited by6

Results whose statement or proof uses this declaration.