🚀 go-pugleaf

I was waiting for some discussion of the following article:

> Article: 351 of sci.math.research
> Subject: Path connectedness
> Message-ID: <1992Sep28.233653.19230@galois.mit.edu>
> From: jbaez@BOURBAKI.MIT.EDU (John C. Baez)
> Date: 28 Sep 92 23:36:53 GMT
> 
> I am posting this for Allen Knutson, aknaton@math.Princeton.EDU,
> who had trouble posting.
> 
> I'm curious about the differences between connectedness and path-connectedness.
> If X is a connected topological space with two marked points, say that Y is
> X-connected if for all y,y' \in Y one can map in X so as to have X's two
> marked points hit y,y' respectively.
> 
> Now define a pre-partial order on connected spaces: "X connects more
> than X'
> does" if being X-connected implies being X'-connected. So for instance, all

Shouldn't it be  "X'-connected implies X-connected"

> R^n's (with whatever two marked points you like) are quickly seen to be
> equivalent, but the topologist's sine curve connects strictly more than [0,1]
> does.
> 
> This partial order (once we mod out by equivalence) plainly has a least
> element: spaces where the two marked points coincide. Does it have any
> maximal elements? Any canonical choice of representative from a given
> connectedness class? Should one restrict to a smaller class of spaces
> (say, compact Hausdorff) for some reason? How tall is it? (If people are
> bugged by the category of connected spaces not being a set, I'm happy
> to just take the ones of cardinality \leq 2^{\aleph_0} or whatever.)
> 
> References would be good too.                                   Allen K.

but since there weren't any contributions, I'll contribute the little I
have.

1) I do not believe the class of compact Hausdorff spaces of cardinality
at most 2^{\aleph_0} has a maximal element under this ordering.  The reason
is the following.  For each ordinal a let X_a be the set of all ordinals
less than a.  On Y_a = (X_a x [0,1)) union ({a} x {0}) take the lexicographic
order, and the order topology.  (The transfinite line [Kelley, General Topology,
Ch. 5, problem L]).  Then the spaces Y_a are compact, Hausdorff,
and connected.  I do not believe there is a compact Hausdorff space of
cardinality at most 2^{\aleph_0} which connects more than all Y_a in
that cardinality.  My feeling is that it should be provable, but I did
not come up with a proof yet.

2) It makes sense that the marked points in the space X be labeled,
and the map should be label preserving.  So, for example, the topologist's
sine curve and the same curve with the labels reversed are incomparable
under this order.

Oded Schramm (schramm@wisdom.weizmann.ac.il)