Connectedness in Topology |
Connected Space > s.a. graph; lie hroup representations.
* Idea: A space which is "all
in one piece"; Of course, this depends crucially on the topology imposed on the set;
Every discrete topological space is "totally" disconnected.
$ Def: A topological space (X, τ)
is connected if the only subsets which are both open and closed are Ø and X.
$ Alternatively: (X, τ)
is connected if there are no non-trivial U, V ∈ τ such
that U ∪ V = X and U ∩ V = Ø.
$ Locally connected space:
For all x ∈ X and neighborhoods U of x,
there is another neighborhood V ⊂ U, with V connected.
* Extremely disconnected space: One in which
every open set has a closure which is open.
* Totally disconnected space:
One in which each connected component is a single point; The only perfect,
totally disconnected metric topological space is the Cantor set, a fractal.
Arcwise (or Pathwise) Connected Space
$ Def: A topological
space (X, τ)
is arcwise connected if for all a, b ∈ X
there is a continuous path q: [0,1] →
X, with q(0) = a and q(1) = b.
* Relationships: Arcwise
connectedness implies connectedness in the usual sense, but not viceversa;
A counterexample is X ∪ Y ⊂ \(\mathbb R\)2,
with X = {(0, x2)
| −1 < x2 < 1}, Y =
{(x1, sin(π/x1)
| 0 < x2 < 1}.
$ Locally: (X, τ)
is locally arcwise connected if for all x ∈ X,
and any neighborhood V(x), ∃ U(x)
⊂ V(x), such that U(x) is arcwise connected.
* Relationships: Local arcwise connectedness implies local connectedness;
There are topological spaces which are simply connected, but not locally pathwise
connected, or not locally connected (think of comb spaces).
@ References: in Singer & Thorpe 67, ch III.
Simply and Multiply Connected Space
$ Simply connected: A
pathwise connected space X with trivial fundamental group, π1(X) = {0}.
$ m-connected: A space X with πp(X)
= {0} for 0 ≤ p ≤ m.
$ Semi-locally simply connected:
A space (X,τ) such that for all x ∈ X
there is a neighborhood U of x such that
any loop in U based at x can be shrunk
to a point in X (not necessarily in U).
* Counterexample: X =
∪n=1∞ Cn,
where Cn is a circle
in \(\mathbb R\)2 with
center at (1/n, 0) and radius 1/n.
Connected Sum of Manifolds or Topological Spaces
> s.a. 3D manifolds; laplacian.
$ Def: In sloppy notation, X # Y:=
(X \ Dn)
∪ (Y \ Dn),
where n is the dimension of X and Y.
* Properties: Associative
and commutative; The identity is Sn.
* Examples:
X # \(\mathbb R\)n = X \ {p}.
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send feedback and suggestions to bombelli at olemiss.edu – modified 16 jan 2016