Connected subsets of topological spaces #
In this file we define connected subsets of a topological spaces and various other properties and classes related to connectivity.
Main definitions #
We define the following properties for sets in a topological space:
IsConnected
: a nonempty set that has no non-trivial open partition. See also the section below in the module doc.connectedComponent
is the connected component of an element in the space.
We also have a class stating that the whole space satisfies that property: ConnectedSpace
On the definition of connected sets/spaces #
In informal mathematics, connected spaces are assumed to be nonempty.
We formalise the predicate without that assumption as IsPreconnected
.
In other words, the only difference is whether the empty space counts as connected.
There are good reasons to consider the empty space to be “too simple to be simple”
See also https://ncatlab.org/nlab/show/too+simple+to+be+simple,
and in particular
https://ncatlab.org/nlab/show/too+simple+to+be+simple#relationship_to_biased_definitions.
A connected set is one that is nonempty and where there is no non-trivial open partition.
Equations
- IsConnected s = (s.Nonempty ∧ IsPreconnected s)
Instances For
If any point of a set is joined to a fixed point by a preconnected subset, then the original set is preconnected as well.
If any two points of a set are contained in a preconnected subset, then the original set is preconnected as well.
A union of a family of preconnected sets with a common point is preconnected as well.
The directed sUnion of a set S of preconnected subsets is preconnected.
The biUnion of a family of preconnected sets is preconnected if the graph determined by whether two sets intersect is preconnected.
The biUnion of a family of preconnected sets is preconnected if the graph determined by whether two sets intersect is preconnected.
Preconnectedness of the iUnion of a family of preconnected sets indexed by the vertices of a preconnected graph, where two vertices are joined when the corresponding sets intersect.
The iUnion of connected sets indexed by a type with an archimedean successor (like ℕ
or ℤ
)
such that any two neighboring sets meet is preconnected.
The iUnion of connected sets indexed by a type with an archimedean successor (like ℕ
or ℤ
)
such that any two neighboring sets meet is connected.
The iUnion of preconnected sets indexed by a subset of a type with an archimedean successor
(like ℕ
or ℤ
) such that any two neighboring sets meet is preconnected.
The iUnion of connected sets indexed by a subset of a type with an archimedean successor
(like ℕ
or ℤ
) such that any two neighboring sets meet is preconnected.
Theorem of bark and tree: if a set is within a preconnected set and its closure, then it is
preconnected as well. See also IsConnected.subset_closure
.
Theorem of bark and tree: if a set is within a connected set and its closure, then it is
connected as well. See also IsPreconnected.subset_closure
.
The closure of a preconnected set is preconnected as well.
The closure of a connected set is connected as well.
The image of a preconnected set is preconnected as well.
The image of a connected set is connected as well.
Alias of Topology.IsInducing.isPreconnected_image
.
If a preconnected set s
intersects an open set u
, and limit points of u
inside s
are
contained in u
, then the whole set s
is contained in u
.
The connected component of a point is the maximal connected set that contains this point.
Equations
- connectedComponent x = ⋃₀ {s : Set α | IsPreconnected s ∧ x ∈ s}
Instances For
Given a set F
in a topological space α
and a point x : α
, the connected
component of x
in F
is the connected component of x
in the subtype F
seen as
a set in α
. This definition does not make sense if x
is not in F
so we return the
empty set in this case.
Equations
- connectedComponentIn F x = if h : x ∈ F then Subtype.val '' connectedComponent ⟨x, h⟩ else ∅
Instances For
A preconnected space is one where there is no non-trivial open partition.
- isPreconnected_univ : IsPreconnected Set.univ
The universal set
Set.univ
in a preconnected space is a preconnected set.
Instances
The universal set Set.univ
in a preconnected space is a preconnected set.
A connected space is a nonempty one where there is no non-trivial open partition.
- isPreconnected_univ : IsPreconnected Set.univ
- toNonempty : Nonempty α
A connected space is nonempty.
Instances
A connected space is nonempty.
Equations
- ⋯ = ⋯
Equations
- ⋯ = ⋯
Equations
- ⋯ = ⋯
Equations
- ⋯ = ⋯
Equations
- ⋯ = ⋯
Equations
- ⋯ = ⋯
Equations
- ⋯ = ⋯