Connectedness
in Topology| 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, 
)
is connected if the only subsets which are both open and closed are Ø and X.
$ Alternatively: (X, )
is connected if there are no nontrivial U, V 
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, )
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 R2,
with X = {(0, x2)
| –1 < x2 < 1}, Y =
{(x1, sin(
/x1)
| 0 < x2 < 1}.
$ Locally: (X, )
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, 1(X)
= {0}.
$ m-connected: A space X with p(X)
= {0} for 0 
p m.
$ Semi-locally simply connected:
A space (X,)
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=1infty Cn,
where Cn is a circle
in R2 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 # Rn = X \ {p}.
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
15 sep 2007