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, *x*_{2})
| −1 < *x*_{2} < 1}, *Y* =
{(*x*_{1}, sin(π/*x*_{1})
| 0 < *x*_{2} < 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, π_{1}(*X*) = {0}.

$ __m____-connected__: A space *X* with π_{p}(*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}^{∞} *C*_{n},
where *C*_{n} 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* \ D^{n})
∪ (*Y* \ D^{n}),
where *n* is the dimension of *X* and *Y*.

* __Properties__: Associative
and commutative; The identity is S^{n}.

* __Examples__:
*X* # \(\mathbb R\)^{n} = *X* \ {*p*}.

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send feedback and suggestions to bombelli at olemiss.edu – modified 16 jan 2016