🚀 go-pugleaf

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
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.