Example interval best-interval-decomposition
[connected] Connected or limit connected := limit point set decomposition is no longer possible or closed set decomposition is no longer possible
with closed ==>
Intuitively speaking, connected = cannot give any substantial decomposition. For any set decomposition , from + connected, each must be connected to some other after the limit point:
Each of the closed set decomposition is an open set
Proof
The definition of connected is equivalent to the version decomposed into two closed sets
with closed ==>
Proof Taking the limit of the decomposition yields
Connected subset := topological subspace connected
Example is connected. has connected and disconnected sets. Connected sets may not be closed sets
[real-connected-is-interval] Connected sets of are intervals Proof by interval connected + optimal interval decomposition + number of intervals in optimal interval decomposition is disconnected
[connected-imply-closure-connected] is a connected set ==> is a connected set
Proof
close ==> closed set is closed set
let closed set decomposition
closed set decomposition and connected ==> one of them is an empty set, say so
But is the smallest closed set containing , so and
is not a connected set ==> is not a connected set
[connected-componet] Connected component decomposition := Limit point set decomposition's limit , such that each limit point set cannot be further decomposed i.e. connected
It is indeed the unique limit in the sense of net. The net comes from the decomposition of into two closed sets, which can be taken as a common refinement decomposition + closed sets are closed under finite intersection.
is connected or cannot be closed set decomposition and has closed set decomposition ==>
Proof The closed set decomposition of , , results in one of the sets being an empty set
is a limit connected set ==> is in the only one limit connected component of
Proof The points of must be in and therefore in some connected component.
==> Even if is only decomposed into connected sets, it is already a connected component decomposition.
The union of connected sets with a common point , , is connected
recall inheritance of subspace topology. So connectedness is also inherited.
So we only need to deal with the case of
Proof The connected sets containing are all in the same connected component. This shows that has only one connected component, and is therefore connected.
A connected component is a maximal element of the maximal linear order of a connected set family.
The image of a continuous function transmits connectedness.
The inverse-image of a continuous function transmits disconnectedness as contrapositive
Proof Closed set decomposition ==> Closed set decomposition
==> [mean-value-theorem-continuous] Intermediate Value Theorem for Continuous Functions. The image of a continuous function is connected therefore is an interval
If any two points in are in some connected subset , then is connected. Proof let with closed, prove that . Or and the union of connected sets that have a common point is connected
==> let be connected. If any two points in are in some connected image of a continuous function, then is connected
==> Path connected
[product-topology-preserve-connected] Product topology preserves connectedness
Proof
Using the common point method + each connected ==> all "cross-shaped" subsets are connected
Using the common point method again, the union of cross-shaped subsets forms a connected subset
and connected-imply-closure-connected ==> connected
Proof of
Just need to prove that each set of the point-net system of intersects some cross shape
The set of the point-net system of is
It intersects the cross shape
let the connected component decomposition
All connected components of are
Proof Using dependent-distributive and the product being connected implies product connectedness, so is connected, thus it can no longer be decomposed
Define (how?) the topology or limit point of (should be something compact open topology? cf. definition of net of analytic-space)
[homotopy] homotopy or limit point homotopy := is limit connected
Example is homotopic to
[homotopy-class] := the connected component of
Since composition preserves continuity, composition induces an operation on . Prove whether it is well-defined. Sometimes it's invertible, making it a group operation