From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: A Problem on Connected Sets
Date: 22 Sep 1999 15:14:50 GMT
Newsgroups: sci.math
Keywords: general remarks about connectivity and topology

In article <37E8DB64.8471AC88@dgsys.com>, Randy Poe  <qnd@dgsys.com> wrote:
>Could you clarify this? What does connected mean exactly as
>opposed to arcwise connected? Is the Cantor set connected?

Topology 101 ...

Definitions:
 Connected -- not the union of two disjoint nonempty open subsets
 Arcwise connected -- for any two points  a, b  there is a continuous  
	f : [0,1] -> X	with f(0)=a, f(1)=b

Theorem: X arcwise connected => X connected

Example: converse does not hold. Consider X = union of two subsets of the plane
	 A = {0} x [-1,1],   B = graph of  y = sin(1/x), x in (0,1]
	 (the "topologists' sine curve")

Theorem: the connected subsets of the real line are precisely the intervals
	 (In particular the Cantor set is not connected)

Remarks: "Connected" is the property more in keeping with the setting of
basic topology: it's about abstract sets with designated subsets called open.
"Arcwise connected" is the property that's more related to the drawing of
pictures, which perhaps makes it more appropriate for geometry. 

Topologists often have this tension regarding the World's Nicest Space, [0,1] :
if you make definitions which refer to it in some way (e.g. CW complexes,
manifolds, metric spaces, ...) you can get stronger results which are more
directly applicable to certain situations which arise outside Topology.
On the other hand, you can't expect the stronger results to apply in
general, and you may have a hard time coming up with "internal" properties
of a topological space (Hausdorff, first countable, etc.) which would 
allow you to draw the desired conclusions.

This is how some of the properties of topological spaces first get
enunciated: one looks to see which properties  [0,1]  has, and which of
those properties are really necessary to make some proof work. Thereafter,
the niceness of a space can be measured by the number of those properties
a space shares with  [0,1]  (connected? compact? normal? ...)

In some sense the great triumphs of topology are those which make the
interval just appear out of nowhere: metrization theorems (If X satisfies
blah blah blah then there is a _metric_ consistent with the topology) and 
Urysohn's lemma (If X is ... there exists a function f : X -> [0,1] with ...)
come to mind.

dave